We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set (PartialConRBDS). Given a bipartite graph G = (R∪ B,E) with red vertices R and blue vertices B, an auxiliary connectivity graph G_{conn} on R, and integers k,t, the task is to find a set S ⊆ R with |S| ≤ k such that G_{conn}[S] is connected and S dominates at least t blue vertices. This formulation captures connected variants of Maximum Coverage [Hochbaum-Rao, Inf. Proc. Lett., 2020; D'Angelo-Delfaraz, AAMAS 2025], Partial Vertex Cover, and Partial Dominating Set [Khuller et al., SODA 2014; Lamprou et al., TCS 2021] via standard encodings. Limits to parameterized tractability. PartialConRBDS is W[1]-hard parameterized by k even under strong restrictions: it remains hard when G_{conn} is a clique or a star and the incidence graph G is 3-degenerate, or when G is K_{2,2}-free. Inapproximability. For every ε > 0, there is no polynomial-time (1, 1-1/e+ε)-approximation unless 𝖯 = NP. Moreover, under ETH, no algorithm running in f(k)⋅ n^{o(k)} time achieves an g(k)-approximation for k for any computable function g(⋅), or for any ε > 0, a (1-1/e+ε)-approximation for t. Graphical special cases. Partial Connected Dominating Set is W[2]-hard parameterized by k and inherits the same ETH-based f(k)⋅ n^{o(k)} inapproximability bound as above; Partial Connected Vertex Cover is W[1]-hard parameterized by k. These hardness boundaries delineate a natural "sweet spot" for study: within appropriate structural restrictions on the incidence graph, one can still aim for fine-grained (FPT) approximations. Our algorithms. We solve PartialConRBDS exactly by reducing it to Relaxed Directed Steiner Out-Tree in time (2e)^t ⋅ n^{𝒪(1)}. For biclique-free incidences (i.e., when G excludes K_{d,d} as an induced subgraph), we obtain two complementary parameterized schemes: - An Efficient Parameterized Approximation Scheme (EPAS) running in time 2^{𝒪(k² d/ε)}⋅ n^{𝒪(1)} that either returns a connected solution of size at most k covering at least (1-ε)t blue vertices, or correctly reports that no connected size-k solution covers t; and - A Parameterized Approximation Scheme (PAS) running in time 2^{𝒪(kd(k²+log d))}⋅ n^{𝒪(1/ε)} that either returns a connected solution of size at most (1+ε)k covering at least t blue vertices, or correctly reports that no connected size-k solution covers t. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

The starting point of our work is the decade-old open question concerning the subexponential parameterized complexity of the 2-Layer Crossing Minimization problem. In this problem, the input is an \(n\)-vertex graph \(G\) whose vertices are divided into two independent sets \(V_1, V_2\), and a non-negative integer \(k\). The question is whether \(G\) supports a 2-layered drawing with at most \(k\) crossings. Here, a 2-layered drawing refers to a drawing of \(G\) where each set \(V_i\) for \(i \in \{1,2\}\) is placed on a distinct straight line parallel to the \(x\)-axis, and all edges are drawn as straight lines connecting vertices. Our first theorem resolves the aforementioned question in the affirmative by providing a fixed-parameter tractable (FPT) subexponential algorithm with running time \(2^{\mathcal O(\sqrt{k}\log k)} + n \cdot k^{\mathcal O(1)}\).

Courcelle’s theorem states that there exists an algorithm that takes as input a graph G of treewidth at most t and a MSO formula φ, and determines whether G satisfies φ in time f(φ,t) · n. It is folklore that the function f contains a tower of exponentials whose height depends as a linear function of the number of quantifier alternations of the input formula φ. A classic reduction of Frick and Grohe shows that, assuming the Exponential Time Hypothesis (ETH), the linear growth of the height of the tower is unavoidable. Nevertheless, there is still a huge gap between existing upper and lower bounds – after all, there is quite a difference between a single exponential and a double exponential running time. In addition, this only gives us a very coarse understanding in the time complexity of Courcelle’s theorem. In this paper, we prove a fine-grained version of Courcelle’s theorem with nearly ETH-tight dependence on the treewidth parameter t and the quantifier structure of φ (specifically, the number of first order and second order variables in each quantifier alternation block).

Abstract Fair division of resources among competing agents is a fundamental problem in computational social choice and game theory. It has been intensively studied for various types of items ( divisible and indivisible ) and under various notions of fairness . We focus on Connected Fair Division ( $$\textsf{CFD}$$ ), the variant of fair division on graphs, where the resources are modeled as an item graph . Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. We introduce a generalization of $$\textsf{CFD}$$ , termed Incomplete $$\textsf{CFD}$$ ( $$\textsf{ICFD}$$ ), where exactly p vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be “economical” as well as fair. We consider four well-known notions of fairness: $$\textsf{PROP}$$ , $$\textsf{EF}$$ , $$\textsf{EF1}$$ , $$\textsf{EFX}$$ . First, we prove that $$\textsf{EF}$$ - $$\textsf{ICFD}$$ , $$\textsf{EF1}$$ - $$\textsf{ICFD}$$ , and $$\textsf{EFX}$$ - $$\textsf{ICFD}$$ are W[1]-hard parameterized by p plus the number of agents, even for graphs having constant vertex cover number ( $$\textsf{vcn}$$ ). In contrast, we present a randomized $$\textsf{FPT}$$ algorithm for $$\textsf{PROP}$$ - $$\textsf{ICFD}$$ parameterized only by p . Additionally, we prove both positive and negative results concerning the kernelization complexity of $$\textsf{ICFD}$$ under all four fairness notions, parameterized by p , $$\textsf{vcn}$$ , and the total number of different valuations in the item graph ( $$\textsf{val}$$ ).

Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber ( CnR ) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber . The aim is to compute the cop number of a graph, k , which is the minimum number of cops that ensures the capture of the robber. From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number ( vcn ). First, we establish that k ≤ vcn 3 + 1 . Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number , and design a linear vertex kernel for neighborhood diversity . Additionally, we extend all of our results to several well-studied variations of CnR .

The class of graph deletion problems has been extensively studied in theoretical computer science, particularly in the field of parameterized complexity. Recently, a new notion of graph deletion problems was introduced, called deletion to scattered graph classes , where after deletion, each connected component of the graph should belong to at least one of the given graph classes. While fixed-parameter algorithms were given for a wide variety of problems, little progress has been made on the kernelization complexity of any of them. Here, we present the first non-trivial polynomial kernel for one such deletion problem, where, after deletion, each connected component should be a clique or a tree - that is, as densest as possible or as sparsest as possible (while being connected). We develop a kernel of O ( k 5 ) vertices for the same.

We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and—in addition to a global budget limit, there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to be hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.

We study the optimization version of the classic Disjoint Paths problem, known as Min-Sum Disjoint Paths , as well as its restriction to shortest paths, known as Disjoint Shortest Paths . Both problems are notoriously hard in the sense that very few positive results are known in their context even when confined to grids, in contrast to the classic Disjoint Paths problem, despite significant research efforts in recent years. In light of this, we focus on restricted graph classes, being subclasses of chordal graphs: specifically, we consider the classes of split graphs, well-partitioned chordal graphs, and threshold graphs. For each of the two problems and each of these graph classes, we provide either a polynomial-time algorithm or a fixed-parameter algorithm (when a polynomial-time algorithm is unlikely to exist).

Knockout tournaments, also known as single-elimination or cup tournaments, are a popular form of sports competitions. In the standard probabilistic setting, for each pairing of players, one of the players wins the game with a certain (a priory known) probability. Due to their competitive nature, tournaments are prone to manipulation. We investigate the computational problem of determining whether, for a given tournament, a coalition has a manipulation strategy that increases the winning probability of a designated player above a given threshold. More precisely, in every round of the tournament, coalition players can strategically decide which games to throw based on the advancement of other players to the current round. We call this setting adaptive constructive coalition manipulation. To the best of our knowledge, while coalition manipulation has been studied in the literature, this is the first work to introduce adaptiveness to this context. We show that the above problem is hard for every complexity class in the polynomial hierarchy. On the algorithmic side, we show that the problem is solvable in polynomial time when the coalition size is a constant. Furthermore, we show that the problem is fixed-parameter tractable when parameterized by the coalition size and the size of a minimum player set that must include at least one player from each non-deterministic game. Lastly, we investigate a generalized setting where the tournament tree can be imbalanced.

Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, T-Cycle: given an undirected n-vertex graph G and a set of k vertices T ⊆ V(G) termed terminals, the objective is to determine whether G contains a simple cycle C through all the terminals. Our contribution is twofold: (i) We provide a 2^{O(√klog k)}⋅ n-time fixed-parameter deterministic algorithm for T-Cycle on planar graphs; (ii) We provide a k^{O(1)}⋅ n-time deterministic kernelization algorithm for T-Cycle on planar graphs where the produced instance is of size klog^{O(1)}k. Both of our algorithms are optimal in terms of both k and n up to (poly)logarithmic factors in k under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for T-Cycle on planar graphs, as well as the first polynomial kernel for T-Cycle on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

We provide the first algorithm for computing an optimal tree decomposition for a given graph $G$ that runs in single exponential time in the feedback vertex number of $G$, that is, in time $2^{O(\text{fvn}(G))}\cdot n^{O(1)}$, where $\text{fvn}(G)$ is the feedback vertex number of $G$ and $n$ is the number of vertices of $G$. On a classification level, this improves the previously known results by Chapelle et al. [Discrete Applied Mathematics '17] and Fomin et al. [Algorithmica '18], who independently showed that an optimal tree decomposition can be computed in single exponential time in the vertex cover number of $G$. One of the biggest open problems in the area of parameterized complexity is whether we can compute an optimal tree decomposition in single exponential time in the treewidth of the input graph. The currently best known algorithm by Korhonen and Lokshtanov [STOC '23] runs in $2^{O(\text{tw}(G)^2)}\cdot n^4$ time, where $\text{tw}(G)$ is the treewidth of $G$. Our algorithm improves upon this result on graphs $G$ where $\text{fvn}(G)\in o(\text{tw}(G)^2)$. On a different note, since $\text{fvn}(G)$ is an upper bound on $\text{tw}(G)$, our algorithm can also be seen either as an important step towards a positive resolution of the above-mentioned open problem, or, if its answer is negative, then a mark of the tractability border of single exponential time algorithms for the computation of treewidth.

Ensemble models are widely recognized in the ML community for their limited interpretability. For instance, while a single decision tree is considered interpretable, ensembles of trees (e.g., boosted trees) are often treated as black-boxes. Despite this folklore recognition, there remains a lack of rigorous mathematical understanding of what particularly makes an ensemble (un)-interpretable, including how fundamental factors like the (1) number, (2) size, and (3) type of base models influence its interpretability. In this work, we seek to bridge this gap by applying concepts from computational complexity theory to study the challenges of generating explanations for various ensemble configurations. Our analysis uncovers nuanced complexity patterns influenced by various factors. For example, we demonstrate that under standard complexity assumptions like P$\neq$NP, interpreting ensembles remains intractable even when base models are of constant size. Surprisingly, the complexity changes drastically with the number of base models: small ensembles of decision trees are efficiently interpretable, whereas ensembles of linear models remain intractable, even with a constant number of models. We believe that our findings provide a more robust foundation for understanding the interpretability of ensembles, emphasizing the benefits of examining it through a computational complexity lens.

The parameterized analysis of graph modification problems represents the most extensively studied area within Parameterized Complexity. Given a graph $G$ and an integer $k\in\mathbb{N}$ as input, the goal is to determine whether we can perform at most $k$ operations on $G$ to transform it into a graph belonging to a specified graph class $\mathcal{F}$. Typical operations are combinatorial and include vertex deletions and edge deletions, insertions, and contractions. However, in many real-world scenarios, when the input graph is constrained to be a geometric intersection graph, the modification of the graph is influenced by changes in the geometric properties of the underlying objects themselves, rather than by combinatorial modifications. It raises the question of whether vertex deletions or adjacency modifications are necessarily the most appropriate modification operations for studying modifications of geometric graphs. We propose the study of the disk intersection graph modification through the scaling of disks. This operation is typical in the realm of topology control but has not yet been explored in the context of Parameterized Complexity. We design parameterized algorithms and kernels for modifying to the most basic graph classes: edgeless, connected, and acyclic. Our technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.

Given a graph G , a matching is a subset of edges of G that do not share an endpoint. A matching M is uniquely restricted if the subgraph induced by the endpoints of the edges of M has exactly one perfect matching. Given a graph G and a positive integer ℓ , Uniquely Restricted Matching asks whether G has a uniquely restricted matching of size at least ℓ . In this paper, we study the parameterized complexity of Uniquely Restricted Matching under various parameters. Specifically, we show that Uniquely Restricted Matching admits a fixed-parameter tractable (FPT) algorithm on line graphs when parameterized by the solution size. We also establish that the problem is FPT when parameterized by the treewidth of the input graph. Furthermore, we show that Uniquely Restricted Matching does not admit a polynomial kernel with respect to the vertex cover number plus the size of the matching unless NP ⊆ coNP/poly.

In this paper, we suggest to extend the notion of a kernel to permit the kernelization algorithm to be executed in quasi-polynomial time rather than polynomial time. So far, we are only aware of one work that addressed this negatively, showing that some lower bounds on kernel sizes proved for kernelization also hold when quasi-polynomial time complexity is allowed. When we, anyway, deal with an NP-hard problem, sacrificing polynomial time in preprocessing for quasi-polynomial time may often not be a big deal, but, of course, the question is - does it give us more power? The only known work, mentioned above, seems to suggest that the answer is "no". In this paper, we show that this is not the case - in particular, we show that this notion is extremely powerful for derandomization. Some of the most basic kernelization algorithms in the field are based on inherently randomized tools whose derandomization is a huge problem that has remained (and may still remain) open for many decades. Still, some breakthrough advances for derandomization in quasi-polynomial time have been made. Can we harness these advancements to design quasi-polynomial deterministic kernelization algorithms for basic problems in the field? To this end, we revisit the question of deterministic polynomial-time computation of a linear representation of transversal matroids and gammoids, which is a longstanding open problem. We present a deterministic computation of a representation matrix of a transversal matroid in time quasipolynomial in the rank of the matroid, where each entry of the matrix can be represented in quasipolynomial (in the rank of the matroid) bits. As a corollary, we obtain a linear representation of a gammoid in deterministic quasipolynomial time and quasipolynomial bits in the size of the underlying ground set of the gammoid. In turn, as applications of our results, we present deterministic quasi-polynomial time kernels of polynomial size for several central problems in the field.

A knockout tournament is one of the most simple and popular forms of competition. Here, we are a given binary tournament tree where all leaves are labeled with seed position names. The players participating in the tournament are assigned to the seed positions. In each round, the two players assigned to leaves of the tournament tree with a common parent compete, and the winner is promoted to the parent. The last remaining player is the winner of the tournament.

A drawing of an (undirected) graph G is a mapping φ that assigns to each vertex a distinct point in the plane and to each edge uw a continuous curve φ (uυ ) in the plane from φ (η ) to φ(ν ), not passing through the image of any other vertex. Two edges e and f cross in a point p if p € φ (e ) ∩ φ (f ) and p is not the image of a vertex of G. In a drawing no three edges are allowed to cross in the same point. The crossing number of a drawing of G is the number of points p such that some two edges e and f cross in p. In the CROSSING Number problem, the input consists of a graph G and integer k. The task is to determine whether there exists a drawing of G with crossing number at most k, and to output such a drawing if it exists.

Challenge the champ tournaments are one of the simplest forms of competition, where a (initially selected) champ is repeatedly challenged by other players. If a player beats the champ, then that player is considered the new (current) champ. Each player in the competition challenges the current champ once in a fixed order. The champ of the last round is considered the winner of the tournament. We investigate a setting where players can be bribed to lower their winning probability against the initial champ. The goal is to maximize the probability of the initial champ winning the tournament by bribing the other players, while not exceeding a given budget for the bribes. Mattei et al. [Journal of Applied Logic, 2015] showed that the problem can be solved in pseudo-polynomial time, and that it is in XP when parameterized by the number of players. We show that the problem is weakly NP-hard and W[1]-hard when parameterized by the number of players. On the algorithmic side, we show that the problem is fixed-parameter tractable when parameterized either by the number of different bribe values or the number of different probability values. To this end, we establish several results that are of independent interest. In particular, we show that the product knapsack problem is W[1]-hard when parameterized by the number of items in the knapsack, and that constructive bribery for cup tournaments is W[1]-hard when parameterized by the number of players. Furthermore, we present a novel way of designing mixed integer linear programs, ensuring optimal solutions where all variables are integers.

Abstract. In the disjoint paths problem, the input is an undirected graph [Formula: see text] on [Formula: see text] vertices and a set of [Formula: see text] vertex pairs, [Formula: see text], and the task is to find [Formula: see text] pairwise vertex-disjoint paths such that the [Formula: see text]’th path connects [Formula: see text] to [Formula: see text]. In this paper, we give a parameterized algorithm with running time [Formula: see text] for planar disjoint paths, the variant of the problem where the input graph is required to be planar. Our algorithm is based on the unique linkage/treewidth reduction theorem for planar graphs by Adler et al. [ J. Combin. Theory Ser. B, 122 (2017), pp. 815–843], the algebraic cohomology based technique of Schrijver [ SIAM J. Comput., 23 (1994), pp. 780–788], and one of the key combinatorial insights developed by Cygan et al. [ Proceedings of the 2013 IEEE 54 th Annual Symposium on Foundations of Computer Science, 2013, pp. 197–206] in their algorithm for disjoint paths on directed planar graphs. To the best of our knowledge, our algorithm is the first parameterized algorithm to exploit the fact that the treewidth of the input graph is small, and it does so in a way that is completely different from the use of dynamic programming.

Abstract. A matching is a subset of edges in a graph [Formula: see text] that do not share an endpoint. A matching [Formula: see text] is a [Formula: see text] -matching if the subgraph of [Formula: see text] induced by the endpoints of the edges of [Formula: see text] satisfies property [Formula: see text]. For example, if the property [Formula: see text] is that of being a matching, being acyclic, or being disconnected, then we obtain an induced matching, an acyclic matching, and a disconnected matching, respectively. Given a graph [Formula: see text] and a positive integer [Formula: see text], the [Formula: see text] Matching problem asks whether [Formula: see text] has a [Formula: see text]-matching of size at least [Formula: see text]. In this paper, we analyze the [Formula: see text] Matching problems from the viewpoint of Parameterized Complexity with respect to the parameter treewidth. In particular, we present a deterministic algorithm solving Induced Matching in [Formula: see text] time and a randomized algorithm solving Acyclic Matching in [Formula: see text] time. For any fixed [Formula: see text], [Formula: see text]-Disconnected Matching can be solved in [Formula: see text] time by a deterministic algorithm. Additionally, assuming the Exponential Time Hypothesis, we show that Disconnected Matching has no [Formula: see text]-time algorithm.

Given a mapping from a set of players to the leaves of a complete binary tree (called a seeding), a knockout tournament is conducted as follows: every round, every two players with a common parent compete against each other, and the winner is promoted to the common parent; then, the leaves are deleted. When only one player remains, it is declared the winner. This is a popular competition format in sports, elections, and decision-making. Over the past decade, it has been studied intensively from both theoretical and practical points of view. Most frequently, the objective is to seed the tournament in a way that ``assists'' (or even guarantees) some particular player to win the competition. We introduce a new objective, which is very sensible from the perspective of the directors of the competition: maximize the profit or popularity of the tournament. Specifically, we associate a ``score'' with every possible match, and aim to seed the tournament to maximize the sum of the scores of the matches that take place. We focus on the case where we assume a total order on the players' strengths, and provide a wide spectrum of results on the computational complexity of the problem.

Decision trees is a fundamental tool in machine learning for representing, classifying, and generalizing data. It is desirable to construct ``small'' decision trees, by minimizing either the size (s) or the depth (d) of the decision tree (DT). Recently, the parameterized complexity of Decision Tree Learning has attracted a lot of attention. We consider a generalization of Decision Tree Learning where given a classification instance E and an integer t, the task is to find a ``small'' DT that disagrees with E in at most t examples. We consider two problems: DTSO and DTDO, where the goal is to construct a DT minimizing s and d, respectively. We first establish that both DTSO and DTDO are W[1]-hard when parameterized by s+y and d+y, respectively, where y is the maximum number of features in which two differently labeled examples can differ. We complement this result by showing that these problems become FPT if we include the parameter t. We also consider the kernelization complexity of these problems and establish several positive and negative results for both DTSO and DTDO.

We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clusetring problem, given a set P of points in R^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L_1-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r=0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given ε>0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+εof the optimum, and it operates in time 2^{(kd/ε)^{O(1)}} n^{O(1)}. Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clusetring.

In a disk graph, every vertex corresponds to a disk in $\mathbb{R}^2$ and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) $3$-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a $(3-α)$-approximation algorithm for Bipartization on disk graphs, for some constant $α>0$. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.

$\newcommand{\eps}{\varepsilon}$ We observe that a $(1-\eps)$-approximation algorithm to Independent Set, that works for any induced subgraph of the input graph, can be used, via a polynomial time reduction, to provide a $(1+\eps)$-approximation to Vertex Cover. This basic observation was made before, see [BHR11]. As a consequence, we get a PTAS for VC for unweighted pseudo-disks, QQPTAS for VC for unweighted axis-aligned rectangles in the plane, and QPTAS for MWVC for weighted polygons in the plane. To the best of our knowledge all these results are new.

We introduce a new framework for the analysis of preprocessing routines for parameterized counting problems. Existing frameworks that encapsulate parameterized counting problems permit the usage of exponential (rather than polynomial) time either explicitly or by implicitly reducing the counting problems to enumeration problems. Thus, our framework is the only one in the spirit of classic kernelization (as well as lossy kernelization). Specifically, we define a compression of a counting problem P into a counting problem Q as a pair of polynomial-time procedures: reduce and lift. Given an instance of P, reduce outputs an instance of Q whose size is bounded by a function f of the parameter, and given the number of solutions to the instance of Q, lift outputs the number of solutions to the instance of P. When P = Q, compression is termed kernelization, and when f is polynomial, compression is termed polynomial compression. Our technical (and other conceptual) contributions can be classified into two categories: Upper Bounds. We prove two theorems: (i) The #Vertex Cover problem parameterized by solution size admits a polynomial kernel; (ii) Every problem in the class of #Planar F-Deletion problems parameterized by solution size admits a polynomial compression. Lower Bounds. We introduce two new concepts of cross-compositions: EXACT-cross-composition and SUM-cross-composition. We prove that if a #P-hard counting problem P EXACT-cross-composes into a parameterized counting problem Q, then Q does not admit a polynomial compression unless the polynomial hierarchy collapses. We conjecture that the same statement holds for SUM-cross-compositions. Then, we prove that: (i) #Min (s,t)-Cut parameterized by treewidth does not admit a polynomial compression unless the polynomial hierarchy collapses; (ii) #Min (s,t)-Cut parameterized by minimum cut size, #Odd Cycle Transversal parameterized by solution size, and #Vertex Cover parameterized by solution size minus maximum matching size, do not admit polynomial compressions unless our conjecture is false.

We study Clustered Planarity with Linear Saturators, which is the problem of augmenting an n-vertex planar graph whose vertices are partitioned into independent sets (called clusters) with paths - one for each cluster - that connect all the vertices in each cluster while maintaining planarity. We show that the problem can be solved in time 2^𝒪(n) for both the variable and fixed embedding case. Moreover, we show that it can be solved in subexponential time 2^𝒪(√n log n) in the fixed embedding case if additionally the input graph is connected. The latter time complexity is tight under the Exponential-Time Hypothesis. We also show that n can be replaced with the vertex cover number of the input graph by providing a linear (resp. polynomial) kernel for the variable-embedding (resp. fixed-embedding) case; these results contrast the NP-hardness of the problem on graphs of bounded treewidth (and even on trees). Finally, we complement known lower bounds for the problem by showing that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most 3, thus excluding the algorithmic use of the number of clusters as a parameter.

The streaming model was introduced to parameterized complexity independently by Fafianie and Kratsch [MFCS14] and by Chitnis, Cormode, Hajiaghayi and Monemizadeh [SODA15]. Subsequently, it was broadened by Chitnis, Cormode, Esfandiari, Hajiaghayi and Monemizadeh [SPAA15] and by Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh and Vorotnikova [SODA16]. Despite its strong motivation, the applicability of the streaming model to central problems in parameterized complexity has remained, for almost a decade, quite limited. Indeed, due to simple Ω(n)-space lower bounds for many of these problems, the kO(1) · polylog(n)-space requirement in the model is too strict.

Abstract The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack $$n+k$$ n + k disks. Thus the problem of packing equal disks is the special case of our problem with $$n=h=0$$ n = h = 0 . While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for $$h=0$$ h = 0 . Our main algorithmic contribution is an algorithm that solves the repacking problem in time $$(h+k)^{\mathcal {O}(h+k)}\cdot |I|^{\mathcal {O}(1)}$$ ( h + k ) O ( h + k ) · | I | O ( 1 ) , where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

We give a new decomposition theorem in unit disk graphs (UDGs) and demonstrate its applicability in the fields of Structural Graph Theory and Parameterized Complexity. First, our new decomposition theorem shows that the class of UDGs admits an “almost” Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010], and H -minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. Even bounded-degree UDGs can contain arbitrarily large cliques as minors, and therefore our result is a significant advance in the study of contraction decompositions. Additionally, this result answers an open question posed by Hajiaghayi ( www.youtube.com/watch?v=2Bq2gy1N01w ) regarding the existence of contraction decompositions for classes of graphs beyond H -minor free graphs though under a relaxation of the original formulation. Second, we present a “parameteric version” of our new decomposition theorem. We prove that there is an algorithm that, given a UDG G and a positive integer k , runs in polynomial time and outputs a collection of \(\mathcal {O}(k)\) tree decompositions of G with the following properties. Each bag in any of these tree decompositions can be partitioned into \(\mathcal {O}(k)\) connected pieces (we call this measure the chunkiness of the tree decomposition). Moreover, for any subset S of at most k edges in G , there is a tree decomposition in the collection such that S is well preserved in the decomposition in the following sense. For any bag in the tree decomposition and any edge in S with both endpoints in the bag, either its endpoints lie in different pieces or they lie in a piece that is a clique. Having this decomposition at hand, we show that the design of parameterized algorithms for some cut problems becomes elementary. In particular, our algorithmic applications include single-exponential (or slightly super-exponential) algorithms for well-studied problems such as Min Bisection , Steiner Cut , s -Way Cut , and Edge Multiway Cut-Uncut on UDGs; these algorithms are substantially faster than the best-known algorithms for these problems on general graphs.

A knockout (or single-elimination) tournament is a format of a competition that is very popular in practice (particularly in sports, elections and decision making), and which has been extensively and intensively studied from a theoretical point of view for more than a decade. Particular attention has been devoted to the Tournament Fixing problem, where, roughly speaking, the objective is to determine whether we can conduct the knockout tournament in a way that makes our favorite player win. Here, part of the input is a tournament graph D that encodes the winner of each possible match. A sequence of papers has studied the parameterized complexity of Tournament Fixing with respect to the feedback arc set number (fas) of D Given that this parameter yielded tractability, it has been asked explicitly and repeatedly whether Tournament Fixing is FPT also with respect to the feedback vertex set number (fvs) of D. We answer this question positively. In fact, although fvs can be arbitrarily smaller than fas, we attain the same dependency on the parameter in the time complexity. So, additionally, our work subsumes the best known algorithm for Tournament Fixing with respect to as.

We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is "close" to each other. More precisely, for a family of n points, an integer k, and a real number d > 0, we ask whether at most k points could be relocated, each point at distance at most d from its original location, such that the distance between each pair of points is at least a fixed constant, say 1. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with 𝒪(d²k³) points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by k and d. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in k alone, unless NP ⊆ coNP/poly.

We re-visit the complexity of polynomial time pre-processing (kernelization) for the d-Hitting Set problem. This is one of the most classic problems in Parameterized Complexity by itself, and, furthermore, it encompasses several other of the most well-studied problems in this field, such as Vertex Cover, Feedback Vertex Set in Tournaments (FVST) and Cluster Vertex Deletion (CVD). In fact, d-Hitting Set encompasses any deletion problem to a hereditary property that can be characterized by a finite set of forbidden induced subgraphs. With respect to bit size, the kernelization complexity of d-Hitting Set is essentially settled: there exists a kernel with 𝒪(k^d) bits (𝒪(k^d) sets and 𝒪(k^{d-1}) elements) and this it tight by the result of Dell and van Melkebeek [STOC 2010, JACM 2014]. Still, the question of whether there exists a kernel for d-Hitting Set with fewer elements has remained one of the most major open problems in Kernelization. In this paper, we first show that if we allow the kernelization to be lossy with a qualitatively better loss than the best possible approximation ratio of polynomial time approximation algorithms, then one can obtain kernels where the number of elements is linear for every fixed d. Further, based on this, we present our main result: we show that there exist approximate Turing kernelizations for d-Hitting Set that even beat the established bit-size lower bounds for exact kernelizations - in fact, we use a constant number of oracle calls, each with "near linear" (𝒪(k^{1+ε})) bit size, that is, almost the best one could hope for. Lastly, for two special cases of implicit 3-Hitting set, namely, FVST and CVD, we obtain the "best of both worlds" type of results - (1+ε)-approximate kernelizations with a linear number of vertices. In terms of size, this substantially improves the exact kernels of Fomin et al. [SODA 2018, TALG 2019], with simpler arguments.

We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC'08]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP.

In the PLANAR DISJOINT PATHS problem, one is given an undirected planar graph with a set of k vertex pairs $\left(s_{i}, t_{i}\right)$ and the task is to find k pairwise vertex-disjoint paths such that the i-th path connects $s_{i}$ to $t_{i}$. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that PLANAR DISJOINT PATHS does not admit a polynomial kernel when parameterized by k unless coNP $\subseteq \mathrm{NP} /$ poly, resolving an open problem by [Bodlaender, Thomassé, Yeo, ESA’09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WKhierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for PLANAR DISJOINT PATHS parameterized by $k+\mathrm{tw}$, where tw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomialtime (Turing) treewidth reduction to $t w=k^{\mathcal{O}(1)}$ under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB’17; Schrijver, SICOMP’94] yields an alternative (and arguably simpler) proof that PLANAR DISJOINT PATHS can be solved in time $2^{\mathcal{O}\left(k^{2}\right)} \cdot n^{\mathcal{O}(1)}$, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC’20].

A temporal graph has an edge set that may change over discrete time steps, and a temporal path (or walk) must traverse edges that appear at increasing time steps. Accordingly, two temporal paths (or walks) are temporally disjoint if they do not visit any vertex at the same time. The study of the computational complexity of finding temporally disjoint paths or walks in temporal graphs has recently been initiated by Klobas et al.. This problem is motivated by applications in multi-agent path finding (MAPF), which include robotics, warehouse management, aircraft management, and traffic routing. We extend Klobas et al.’s research by providing parameterized hardness results for very restricted cases, with a focus on structural parameters of the so-called underlying graph. On the positive side, we identify sufficiently simple cases where we can solve the problem efficiently. Our results reveal some surprising differences between the “path version” and the “walk version” (where vertices may be visited multiple times) of the problem, and answer several open questions posed by Klobas et al.

Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set 𝒮 of r-dimensional simplices of minimum cardinality so that 𝒮 meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+Δ, where Δ is the maximum degree of the Hasse graph of the complex 𝖪.

We initiate the study of the parameterized complexity of the Collective Graph Exploration (CGE) problem. In CGE, the input consists of an undirected connected graph G and a collection of k robots, initially placed at the same vertex r of G, and each one of them has an energy budget of B. The objective is to decide whether G can be explored by the k robots in B time steps, i.e., there exist k closed walks in G, one corresponding to each robot, such that every edge is covered by at least one walk, every walk starts and ends at the vertex r, and the maximum length of any walk is at most B. Unfortunately, this problem is NP-hard even on trees [Fraigniaud et al., 2006]. Further, we prove that the problem remains W[1]-hard parameterized by k even for trees of treedepth 3. Due to the para-NP-hardness of the problem parameterized by treedepth, and motivated by real-world scenarios, we study the parameterized complexity of the problem parameterized by the vertex cover number (vc) of the graph, and prove that the problem is fixed-parameter tractable (FPT) parameterized by vc. Additionally, we study the optimization version of CGE, where we want to optimize B, and design an approximation algorithm with an additive approximation factor of O(vc).

Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such algorithms. An even more serious issue is that, here, "standard" parameters very often yield intractability. In particular, for the most common structural parameter, namely, treewidth, we frequently observe NP-hardness already when the input graphs are restricted to have constant (often, being just $1$ or $2$) treewidth. Our work deals with both drawbacks simultaneously. We introduce a novel form of tree decomposition that, roughly speaking, does not decompose (only) a graph, but an entire drawing. As such, its bags and separators are of geometric (rather than only combinatorial) nature. While the corresponding parameter -- like treewidth -- can be arbitrarily smaller than the height (and width) of the drawing, we show that -- unlike treewidth -- it gives rise to efficient algorithms. Specifically, we get slice-wise polynomial (XP) time algorithms parameterized by our parameter. We present a general scheme for the design of such algorithms, and apply it to several central problems in Graph Drawing, including the recognition of grid graphs, minimization of crossings and bends, and compaction. Other than for the class of problems we discussed in the paper, we believe that our decomposition and scheme are of independent interest and can be further extended or generalized to suit even a wider class of problems. Additionally, we discuss classes of drawings where our parameter is bounded by $O(\sqrt{n})$ (where $n$ is the number of vertices of the graph), yielding subexponential-time algorithms. Lastly, we prove which relations exist between drawn treewidth and other width measures, including treewidth, pathwidth, (dual) carving-width and embedded-width.

We study the classic Dominating Set problem with respect to several prominent parameters. Specifically, we present algorithmic results that sidestep time complexity barriers by the incorporation of either approximation or larger parameterization. Our results span several parameterization regimes, including: (i,ii,iii) time/ratio-tradeoff for the parameters treewidth, vertex modulator to constant treewidth and solution size; (iv,v) FPT-algorithms for the parameters vertex cover number and feedback edge set number; and (vi) compression for the parameter feedback edge set number.

We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex- deletion problems on disk graphs, which results in EPTASes for many fundamental problems including VERTEX COVER, FEEDBACK VERTEX SET, SMALL CYCLE HITTING (in particular, TRIANGLE HITTING), Pk-VERTEX DELETION for k ∈ {3,4,5}, PATH DELETION, PATHWIDTH 1-DELETION, COMPONENT ORDER CONNECTIVITY, BOUNDED DEGREE DELETION, PSEUDOFOREST DELETION, FINITE-TYPE COMPONENT DELETION, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input disk graph (in fact, we allow the input to be any graph, and our algorithms either output a correct approximation solution for the problem or conclude that the input graph is not a disk graph). To the best of our knowledge, prior to this work, the only problems known to admit PTASes or EPTASes on disk graphs are MAXIMUM CLIQUE, INDEPENDENT SET, DOMINATING SET, and VERTEX COVER, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for VERTEX COVER require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a “mild” generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius. * The full version of the paper can be accessed at https://arxiv.org/abs/2211.02717

Abstract Fradkin and Seymour (J Comb Theory Ser B 110:19–46, 2015) defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms (a trait uncommon to parameterized algorithms) on tournaments, including Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk (in: Proceedings of the Algorithms—ESA 2013—21st Annual European Symposium, Sophia Antipolis, France, September 2–4, 2013, pp. 505–516, 2013), where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is a combinatorial result that proves that the yes-instances of the problems (defined above) have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, inductive reasoning and exploiting the structural properties of these digraphs.

Given a graph G and an integer k , the Interval Vertex Deletion (IVD) problem asks whether there exists a subset S ⊆ V ( G ) of size at most k such that G-S is an interval graph. This problem is known to be NP -complete (according to Yannakakis at STOC 1978). Originally in 2012, Cao and Marx showed that IVD is fixed parameter tractable: they exhibited an algorithm with running time 10 k n O (1). The existence of a polynomial kernel for IVD remained a well-known open problem in parameterized complexity. In this article, we settle this problem in the affirmative.

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack n+k disks. Thus the problem of packing equal disks is the special case of our problem with n = h = 0. While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for h = 0. Our main algorithmic contribution is an algorithm that solves the repacking problem in time (h+k)^𝒪(h+k)⋅|I|^𝒪(1), where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

The Delaunay graph of a point set P ⊆ ℝ² is the plane graph with the vertex-set P and the edge-set that contains {p,p'} if there exists a disc whose intersection with P is exactly {p,p'}. Accordingly, a triangulated graph G is Delaunay realizable if there exists a triangulation of the Delaunay graph of some P ⊆ ℝ², called a Delaunay triangulation of P, that is isomorphic to G. The objective of Delaunay Realization is to compute a point set P ⊆ ℝ² that realizes a given graph G (if such a P exists). Known algorithms do not solve Delaunay Realization as they are non-constructive. Obtaining a constructive algorithm for Delaunay Realization was mentioned as an open problem by Hiroshima et al. [Hiroshima et al., 2000]. We design an n^𝒪(n)-time constructive algorithm for Delaunay Realization. In fact, our algorithm outputs sets of points with integer coordinates.

Vertex-deletion problems have been at the heart of parameterized complexity throughout its history. Here, the aim is to determine the minimum size (denoted by modℋ) of a modulator to a graph class ℋ, i.e., a set of vertices whose deletion results in a graph in ℋ. Recent years have seen the development of a research programme where the complexity of modulators is measured in ways other than size. For instance, for a graph class ℋ, the graph parameters elimination distance to ℋ (denoted by edℋ) [Bulian and Dawar, Algorithmica, 2016] and ℋ-treewidth (denoted by twℋ) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the “torso” of the graph induced on a modulator to the graph class ℋ. Here, the torso of a vertex set S in a graph G is the graph with vertex set S and an edge between two vertices u, v ∊ S if there is a path between u and v in G whose internal vertices all lie outside S. In this paper, we show that from the perspective of (non-uniform) fixed-parameter tractability (FPT), the three parameters described above give equally powerful parameterizations for every hereditary graph class ℋ that satisfies mild additional conditions. In fact, we show that for every hereditary graph class ℋ satisfying mild additional conditions, with the exception of edℋ parameterized by twℋ, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to ℋ) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing edℋ and twℋ. The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if ℋ is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a “strong irrelevant vertex rule”, then there exists a uniform FPT algorithm for edℋ. Using these sufficient conditions, we obtain uniform FPT algorithms for computing edℋ, when ℋ is defined by excluding a finite number of connected (a) minors, or (b) topological minors, or (c) induced subgraphs, or when ℋ is any of bipartite, chordal or interval graphs. For most of these problems, the existence of a uniform FPT algorithm has remained open in the literature. In fact, for some of them, even a non-uniform FPT algorithm was not known. For example, Jansen et al. [STOC 2021] ask for such an algorithm when ℋ is defined by excluding a finite number of connected topological minors. We resolve their question in the affirmative.

One of the most celebrated results in Parameterized Complexity is the Bidimensionality theory of Demaine et al. [J. ACM, 2005], which has yielded, over the past two decades, numerous subexponential-time fixed-parameter tractable (FPT) algorithms for various problems on planar (and H-minor-free) graphs. At the heart of this theory is the proof of sublinear bounds in terms of solution size on the treewidth of a given graph. Inspired by this theory, in recent years, significant efforts have been devoted to design subexponential-time FPT algorithms for problems on geometric graph classes that utilize new treewidth bounds, in particular (but not only) for unit disk graphs [Fomin et al., SODA'12; Fomin et al., DCG'19; Panolan et al., SODA'19; Fomin et al. SoCG'20]. In this paper, we aim to attain such results on disk graphs, a broad class of graphs that generalizes both the classes of planar graphs and unit disk graphs, and thereby unify the aforementioned research frontiers for planar and unit disk graphs. Our main contribution is an approach to design subexponential-time FPT algorithms for problems on disk graphs, which we apply to several well-studied graph problems. At the heart of our approach lie two new combinatorial theorems concerning the treewidth of disk graphs having a realization of bounded ply (or maximum clique size) that are of independent interest. In particular, we prove a stronger version of the following treewidth bound: Let G be a disk graph that has some realization of ply p and no false twins, and M ⊆ V(G) such that G has no triangle with exactly one vertex from M, and G–M has treewidth w. Then, the treewidth of G is . Among our applications are the first subexponential-time FPT algorithms for several problems on disk graphs, including Triangle Hitting, Feedback Vertex Set and Odd Cycle Transversal (OCT). Previously, subexponential-time FPT algorithms for these problems were only known on planar graphs and unit disk graphs (excluding OCT, which was only known to admit such an algorithm on planar graphs). Our algorithms are robust, in particular, they do not require a geometric realization of the input graph (for all aforementioned problems), and they generalize to the weighted and counting versions of all aforementioned problems except for OCT.

The Terrain Guarding problem is a well-known variant of the famous Art Gallery problem. Only second to Art Gallery , it is the most well-studied visibility problem in Discrete and Computational Geometry, which has also attracted attention from the viewpoint of Parameterized complexity. In this paper, we focus on the parameterized complexity of Terrain Guarding (both discrete and continuous) with respect to two natural parameters. First we show that, when parameterized by the number r of reflex vertices in the input terrain, the problem has a polynomial kernel. We also show that, when parameterized by the number c of minima in the terrain, Discrete Orthogonal Terrain Guarding has an XP algorithm.

Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESMI), Balanced Stable Marriage (BSMI), max-Stable Marriage with Ties (max-SMTI), and min-Stable Marriage with Ties (min-SMTI) problems. In this paper, we analyze these problems from the viewpoint of parameterized complexity. We conduct the first study of these problems in particular, and of problems related to Stable Marriage in general, with respect to the parameter treewidth. The motivation behind the choice of treewidth is threefold. First, several problems in social choice theory have already been studied with respect to treewidth. The networks relevant to these problems (say, social networks) are clearly also relevant to Stable Marriage. Thus, the motivation underlying these studies directly extends to our study. Second, empirical studies of the treewidth of several types of networks relevant to Stable Marriage have also already been undertaken, identifying that some of these networks indeed have a treelike structure. Third, treewidth is the most well studied structural parameter in parameterized complexity. We design optimal parameterized algorithms for all four problems under the treewidth of both their primal graphs and rotation digraphs. First, we study the treewidth ${\mathtt{tw}}$ of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time $n^{{\mathcal{O}}({\mathtt{tw}})}$, we prove that unless the exponential-time hypothesis is false, all of these algorithms are optimal. Next, we study the treewidth ${\mathtt{tw}}$ of the rotation digraph. In this context, max-SMTI and min-SMTI are not defined. For both SESMI and BSMI, we design (highly nontrivial) algorithms that run in time $2^{{\mathtt{tw}}}n^{{\mathcal{O}}(1)}$. Then, for both SESMI and BSMI, we prove that unless the strong exponential-time hypothesis is false, algorithms that run in time $(2-\epsilon)^{{\mathtt{tw}}}n^{{\mathcal{O}}(1)}$ do not exist for any fixed $\epsilon>0$. We thus present a comprehensive, complete picture of the behavior of Stable Marriage with respect to treewidth.

We study a set of voting problems where given an election E=(C,ΠV) (where C is the set of candidates and ΠV is a set of votes), and a non-empty subset of candidates J, the question under consideration is: Can we modify the election in a way so that none of the candidates in J wins the election? The modification operations allowed are that of either adding or deleting some candidates. Yang and Wang (2017) [44] introduced these problems as the Resolute Control problem, a generalization of the destructive control problem where J is a singleton. They studied parameterized complexity of Resolute Control for voting rules Borda (both addition and deletion), Maximin (addition), and Copeland (both addition and deletion). They primarily consider |J| as parameter. In this paper we study Resolute Control parameterized by the other natural parameters viz., the number of candidates added or deleted. We show that the Resolute Control for Borda (both addition and deletion), Maximin (addition) and Copeland (deletion) are W[2]-hard. We complement this by showing that when the number of voters is odd, Copeland (deletion) is FPT parameterized by the sum of the number of deleted candidates and the size of the feedback arc set of the majority graph of the election.

We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and---in addition to a global budget limit---there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to being hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.

Grid graphs, and, more generally, k×r grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph (given a graph G, decide whether it can be embedded into a grid graph) is particularly hard - it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present new and complementary hardness results). Specifically, our contribution is threefold. First, we show that the problem is fixed-parameter tractable (FPT) parameterized by k+mcc where mcc is the maximum size of a connected component of G. This also implies that the problem is FPT parameterized by td+k where td is the treedepth of G, as td ≤ mcc (to be compared with the hardness for pathwidth 2 where k = 3). (We note that when k and r are unrestricted, the problem is trivially FPT parameterized by td.) Further, we derive as a corollary that strip packing is FPT with respect to the height of the strip plus the maximum of the dimensions of the packed rectangles, which was previously only known to be in XP. Second, we present a new parameterization, denoted a_G, relating graph distance to geometric distance, which may be of independent interest. We show that the problem is para-NP-hard parameterized by a_G, but FPT parameterized by a_G on trees, as well as FPT parameterized by k+a_G. Third, we show that the recognition of k× r grid graphs is NP-hard on graphs of pathwidth 2 where k = 3. Moreover, when k and r are unrestricted, we show that the problem is NP-hard on trees of pathwidth 2, but trivially solvable in polynomial time on graphs of pathwidth 1.

Over the past decade, many results have focused on the design of parameterized approximation algorithms for W[1]-hard problems. However, there are fundamental problems within the class FPT for which the best known algorithms have seen no progress over the course of the decade; some of them have even been proved not to admit algorithms that run in time 2(k) n(1) under the Exponential Time Hypothesis (ETH) or (c – ∊)k n(1) under the Strong ETH (SETH). In this paper, we expand the study of FPT-approximation and initiate a systematic study of FPT-approximation for problems that are FPT. We design FPT-approximation algorithms for problems that are FPT, with running times that are significantly faster than the corresponding best known FPT-algorithm, and while achieving approximation ratios that are significantly better than what is possible in polynomial time. We present a general scheme to design 2(k) n(1)-time 2-approximation algorithms for cut problems. In particular, we exemplify it for Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. Further, we extend our scheme to obtain FPT-time (1)-approximation algorithms for weighted cut problems, where the objective is to obtain a solution of size at most k and of minimum weight. Here, we present two approaches. The first approach achieves 2(k) n(1)-time constant-factor approximation, which we exemplify for all problems mentioned in the first bullet. The other leads to an FPT-approximation Scheme (FPT-AS) for Weighted Directed Feedback Vertex Set. Additionally, we present a combinatorial lemma that yields a partition of the vertex set of a graph to roughly equal sized sets so that the removal of each set reduces its treewidth substantially, which may be of independent interest. For several graph problems, use this lemma to design cwn(1)-time (1 + ∊)-approximation algorithms that are faster than known SETH lower bounds, where w is the treewidth of the input graph. Examples of such problems include Vertex Cover, Component Order Connectivity, Bounded-Degree Vertex Deletion and ℱ-Packing for any family ℱ of bounded sized graphs. Lastly, we present a general reduction of problems parameterized by treewidth to their versions parameterized by solution size. Combined with our first scheme, we exemplify it to obtain cwn(1)-time bicriteria approximation algorithms for all problems mentioned in the first bullet.

In this paper we prove an analogue of the classic Bollobás lemma for approximate counting. In fact, we match an analogous result of Fomin et al. [JACM 2016] for decision. This immediately yields, for a number of fundamental problems, parameterized approximate counting algorithms with the same running times as what is obtained for the decision variant using the representative family technique of Fomin et al. [JACM 2016]. For example, we devise an algorithm for approximately counting (a factor (1 ± ∊) approximation algorithm) k-paths in an n-vertex directed graph (#k-Path) running in time (n + m)). This improves over an earlier algorithm of Brand et al. [STOC 2018] that runs in time . Additionally, we obtain an approximate counting analogue of the efficient computation of representative families for product families of Fomin et al. [TALG 2017], again essentially matching the running time for decision. This results in an algorithm with running time for computing a (1 + ∊) approximation of the sum of the coefficients of the multilinear monomials in a degree-k homogeneous n-variate polynomial encoded by a monotone circuit (#Multilinear Monomial Detection). When restricted to monotone circuits (rather than polynomials of non-negative coefficients), this improves upon an earlier algorithm of Pratt [FOCS 2019] that runs in time .

Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most Ω(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized.

Abasi et al. (2014) introduced the following two problems. In the r -S imple k -P ath problem, given a digraph G on n vertices and positive integers r , k , decide whether G has an r -simple k -path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k . In the ( r , k )-M onomial D etection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k , r , decide whether P has a monomial of total degree k where the degree of each variable is at most r . Abasi et al. obtained randomized algorithms of running time 4 ( k / r )log r ⋅ n O (1) for both problems. Gabizon et al. (2015) designed deterministic 2 O (( k / r )log r ) ⋅ n O (1) -time algorithms for both problems (however, for the ( r , k )-M onomial D etection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the P -S et ( r , q )-P acking P roblem , given a universe V , positive integers ( p , q , r ), and a collection H of sets of size P whose elements belong to V , decide whether there exists a subcollection H ′ of H of size q where each element occurs in at most r sets of H ′ . Gabizon et al. obtained a deterministic 2 O (( pq / r )log r ) ⋅ n O (1) -time algorithm for P -S et ( r , q )-P acking . The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) parameterized by the product of two parameters, that is, k / r and log r , where k = pq for P -S et ( r , q )-P acking . Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for ( r , k )-M onomial D etection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2 o (( k / r ) log r ) ⋅ ( n + log k ) O (1) -time algorithm for ( r , k )-M onomial D etection , i.e., ( r , k )-M onomial D etection is unlikely to be single-exponentially FPT when parameterized by k / r alone. The question remains open for r -S imple k -P ath and P -S et ( r , q )-P acking . We consider the question from a wider perspective: are the above problems FPT when parameterized by k / r only, i.e., whether there exists a computable function f such that the problems admit a f ( k / r )( n +log k ) O (1) -time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k / r alone. We resolve the wider question by (a) obtaining a 2 O (( k / r ) 2 log( k / r )) ⋅ ( n + log k ) O (1) -time algorithm for

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).

The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be ∃ℝ-complete [Abrahamsen et al., STOC'18]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time r^O(r²)n^O(1). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{𝒪(√k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√k)}(n+m)^𝒪(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{𝒪(√k)}(n+m)^𝒪(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{𝒪(√klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width 𝒪(√k).

In the Stable Marriage problem, when the preference lists are complete, all agents of the smaller side can be matched. However, this need not be true when preference lists are incomplete. In most real-life situations, where agents participate in the matching market voluntarily and submit their preferences, it is natural to assume that each agent wants to be matched to someone in his/her preference list as opposed to being unmatched. In light of the Rural Hospital Theorem, we have to relax the "no blocking pair" condition for stable matchings in order to match more agents. In this paper, we study the question of matching more agents with fewest possible blocking edges. In particular, the goal is to find a matching whose size exceeds that of a stable matching in the graph by at least t and has at most k blocking edges. We study this question in the realm of parameterized complexity with respect to several natural parameters, k,t,d, where d is the maximum length of a preference list. Unfortunately, the problem remains intractable even for the combined parameter k+t+d. Thus, we extend our study to the local search variant of this problem, in which we search for a matching that not only fulfills each of the above conditions but is "closest", in terms of its symmetric difference to the given stable matching, and obtain an FPT algorithm.

We prove that the Hadwiger number of an $n$-vertex graph $G$ (the maximum size of a clique minor in $G$) cannot be computed in time $n^{o(n)}$, unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of $n^{o(n)}$-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs.

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ⋅ g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε>0, multiplicative parameterization above g(I)^(1+ε) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth.

In the past decade, the design of fault tolerant data structures for networks has become a central topic of research. Particular attention has been given to the construction of a subgraph H of a given digraph D with as fewest arcs/vertices as possible such that, after the failure of any set F of at most k ≥ 1 arcs, testing whether D-F has a certain property P is equivalent to testing whether H-F has that property. Here, reachability (or, more generally, distance preservation) is the most basic requirement to maintain to ensure that the network functions properly. Given a vertex s ∈ V(D), Baswana et al. [STOC'16] presented a construction of H with O(2^kn) arcs in time O(2^{k}nm) where n=|V(D)| and m= |E(D)| such that for any vertex v ∈ V(D): if there exists a path from s to v in D-F, then there also exists a path from s to v in H-F. Additionally, they gave a tight matching lower bound. While the question of the improvement of the dependency on k arises for special classes of digraphs, an arguably more basic research direction concerns the dependency on n (for reachability between a pair of vertices s,t ∈ V(D)) - which are the largest classes of digraphs where the dependency on n can be made sublinear, logarithmic or even constant? Already for the simple classes of directed paths and tournaments, Ω(n) arcs are mandatory. Nevertheless, we prove that "almost acyclicity" suffices to eliminate the dependency on n entirely for a broad class of dense digraphs called bounded independence digraphs. Also, the dependence in k is only a polynomial factor for this class of digraphs. In fact, our sparsification procedure extends to preserve parity-based reachability. Additionally, it finds notable applications in Kernelization: we prove that the classic Directed Feedback Arc Set (DFAS) problem as well as Directed Edge Odd Cycle Transversal (DEOCT) (which, in sharp contrast to DFAS, is W[1]-hard on general digraphs) admit polynomial kernels on bounded independence digraphs. In fact, for any p ∈ N, we can design a polynomial kernel for the problem of hitting all cycles of length ℓ where (ℓ mod p = 1). As a complementary result, we prove that DEOCT is NP-hard on tournaments by establishing a combinatorial identity between the minimum size of a feedback arc set and the minimum size of an edge odd cycle transversal. In passing, we also improve upon the running time of the sub-exponential FPT algorithm for DFAS in digraphs of bounded independence number given by Misra et at. [FSTTCS 2018], and give the first sub-exponential FPT algorithm for DEOCT in digraphs of bounded independence number.

In this paper, we prove a new scaling lemma for vertex weighted minor free graphs that allows for a smooth trade-off between the weight of a vertex set S and the treewidth of G — S. More precisely, we show the following. There exists an algorithm that given an H-minor free graph G, a weight function w: V(G) → ℚ+ and integers t and s, runs in polynomial time, and outputs a subset S ⊆ V(G) of weight at most d log n · opt(G, w, t)/s such that the treewidth of G – S is at most c·st. Here, d and c are fixed constants that depend only on H, and opt(G, w, t) is the (unknown) minimum weight of a subset U ⊆ V(G) such that the treewidth of G – U is at most t. This lemma immediately yields the first polynomial-time approximation schemes (PTASes) for WEIGHTED Treewidth-η Vertex Deletion, for η > 2, on graphs of bounded genus and the first PTAS for Weighted Feedback vertex Set on H-minor free graphs. These results effortlessly generalize to include weighted edge deletion problems, to all Weighted Connected Planar -Deletion problems, and finally to quasi polynomial time approximation schemes (QPTASes) for all of these problems on H-minor free graphs. For most of these problems even constant factor approximation algorithms, even on planar graphs, were not previously known. Additionally, using the scaling lemma we subsume, simplify and extend the recent framework of Cohen-Addad et al. [STOC 2016] for turning constant factor approximation algorithms for “ubiquitous” problems into PTASes for the same problems on graphs of bounded genus. Specifically, we obtain PTASes for ubiquitous problems without the requirement of having a constant factor approximation. While the statement of the scaling lemma is inspired by an analogous lemma by Cohen-Addad et al. [STOC 2016] for edge contractions on weighted graphs of bounded genus, as well as a scaling lemma by Fomin et al. [SODA 2011] for unweighted graphs, the proof is entirely different. The proof detours via three different linear programming relaxations for the Weighted Treewidth-η Vertex Deletion problems and a strengthening of a recent rounding procedure of Bansal et al. [SODA 2017] enhanced by the classic Klein-Plotkin-Rao Theorem [STOC 1993].

A directed odd cycle transversal of a directed graph (digraph) D is a vertex set S that intersects every odd directed cycle of D. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph D and an integer k. The objective is to determine whether there exists a directed odd cycle transversal of D of size at most k. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size k by showing that DOCT does not admit an algorithm with running time unless FPT = W[1]. On the positive side, we give a factor 2 fixed-parameter approximation (FPT approximation) algorithm for the problem. More precisely, our algorithm takes as input D and k, runs in time , and either concludes that D does not have a directed odd cycle transversal of size at most k, or produces a solution of size at most 2k. Finally, assuming gap-ETH, we show that there exists an ϵ > 0 such that DOCT does not admit a factor (1 + ϵ) FPT-approximation algorithm.

In the Topological Minor Deletion (TM-Deletion) problem, the input consists of an undirected graph G, a family of undirected graphs F and an integer k. The task is to determine whether G contains a set of vertices S of size at most k, such that the graph G∖ S obtained from G by removing the vertices of S, contains no graph from F as a topological minor. We give an algorithm forTM-Deletion with running time f(h ⋆,k)· |V(G)|4. Here h ⋆ is the maximum size of a graph in F and f is a computable function of h ⋆ and k. This is the first fixed parameter tractable algorithm (FPT) for the problem. In fact, even for the restricted case of planar inputs the first FPT algorithm was found only recently by Golovach et al. [SODA 2020]. For this case we improve upon the algorithm of Golovach et al. [SODA 2020] by designing an FPT algorithm with explicit dependence on k and h ⋆.

In the Disjoint Paths problem, the input is an undirected graph G on n vertices and a set of k vertex pairs, {s i ,t i } i=1 k , and the task is to find k pairwise vertex-disjoint paths such that the i’th path connects s i to t i . In this paper, we give a parameterized algorithm with running time 2 O(k 2) n O(1) for Planar Disjoint Paths, the variant of the problem where the input graph is required to be planar. Our algorithm is based on the unique linkage/treewidth reduction theorem for planar graphs by Adler et al. [JCTB 2017], the algebraic co-homology based technique developed by Schrijver [SICOMP 1994] for Disjoint Paths on directed planar graphs, and one of the key combinatorial insights developed by Cygan et al. [FOCS 2013] in their algorithm for Disjoint Paths on directed planar graphs. To the best of our knowledge our algorithm is the first parameterized algorithm to exploit that the treewidth of the input graph is small in a way completely different from the use of dynamic programming.

We define a new problem in comparative genomics, denoted PQ-Tree Search, that takes as input a PQ-tree $T$ representing the known gene orders of a gene cluster of interest, a gene-to-gene substitution scoring function $h$, integer parameters $d_T$ and $d_S$, and a new genome $S$. The objective is to identify in $S$ approximate new instances of the gene cluster that could vary from the known gene orders by genome rearrangements that are constrained by $T$, by gene substitutions that are governed by $h$, and by gene deletions and insertions that are bounded from above by $d_T$ and $d_S$, respectively. We prove that the PQ-Tree Search problem is NP-hard and propose a parameterized algorithm that solves the optimization variant of PQ-Tree Search in $O^*(2^γ)$ time, where $γ$ is the maximum degree of a node in $T$ and $O^*$ is used to hide factors polynomial in the input size. The algorithm is implemented as a search tool, denoted PQFinder, and applied to search for instances of chromosomal gene clusters in plasmids, within a dataset of 1,487 prokaryotic genomes. We report on 29 chromosomal gene clusters that are rearranged in plasmids, where the rearrangements are guided by the corresponding PQ-tree. One of these results, coding for a heavy metal efflux pump, is further analysed to exemplify how PQFinder can be harnessed to reveal interesting new structural variants of known gene clusters. The code for the tool as well as all the data needed to reconstruct the results are publicly available on GitHub (github.com/GaliaZim/PQFinder).

We study a motion-planning problem inspired by the game Snake that models scenarios like the transportation of linked wagons towed by a locomotor to the movement of a group of agents that travel in an ``ant-like'' fashion. Given a ``snake-like'' robot with initial and final positions in an environment modeled by a graph, our goal is to decide whether the robot can reach the final position from the initial position without intersecting itself. Already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2018]. Nevertheless, we prove that even on general graphs, it is solvable in time k^{O(k)}|I|^{O(1)} where k is the size of the robot, and |I| is the input size. Towards this, we give a novel application of color-coding to sparsify the configuration graph of the problem. We also show that the problem is unlikely to have a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion problems has been~largely~neglected, thus our work is pioneering in this regard.

An undirected graph $G$ is $d$-degenerate if every subgraph of $G$ has a vertex of degree at most $d$. By the classical theorem of Erdös and Gallai from 1959, every graph of degeneracy $d>1$ contains a cycle of length at least $d+1$. The proof of Erdös and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least $d+1$. But can we decide in polynomial time whether a graph contains a cycle of length at least $d+2$? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least $d+2$ is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether $G$ contains a cycle of length at least $d+k$ can be done in time $2^{\mathcal{O}(k)}\cdot|V(G)|^{\mathcal{O}(1)}$. In other words, deciding whether a 2-connected $n$-vertex $G$ contains a cycle of length at least $d+\log{n}$ can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least $d+1$ is NP-complete. However, we prove that if graph $G$ is connected, then deciding whether $G$ contains a path of length at least $d+k$ can be done in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$. We complement these results by showing that the choice of degeneracy as the “above guarantee parameterization” is optimal in the following sense: For any $\varepsilon>0$ it is NP-complete to decide whether a connected (2-connected) graph of degeneracy $d$ has a path (cycle) of length at least $(1+\varepsilon)d$.

For a family of graphs ℱ, the W eighted ℱ V ertex D eletion problem, is defined as follows: given an n -vertex undirected graph G and a weight function w : V ( G )࢐ ℝ, find a minimum weight subset S ⊆ V ( G ) such that G - S belongs to ℱ. We devise a recursive scheme to obtain O(log O(1) n )-approximation algorithms for such problems, building upon the classical technique of finding balanced separators . We obtain the first O(log O(1) n )-approximation algorithms for the following problems. • Let F be a finite set of graphs containing a planar graph, and ℱ= G ( F ) be the maximal family of graphs such that every graph H ∈ G ( F ) excludes all graphs in F as minors. The vertex deletion problem corresponding to ℱ= G ( F ) is the W eighted P lanar F -M inor -F ree D eletion (WP F -MFD) problem. We give a randomized and a deterministic approximation algorithms for WP F -MFD with ratios O(log 1.5 n ) and O(log 2 n ), respectively. Prior to our work, a randomized constant factor approximation algorithm for the unweighted version was known [FOCS 2012]. After our work, a deterministic constant factor approximation algorithm for the unweighted version was also obtained [SODA 2019]. • We give an O(log 2 n )-factor approximation algorithm for W eighted C hordal V ertex D eletion , the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for M ulticut on chordal graphs. • We give an O(log 3 n )-factor approximation algorithm for W eighted D istance H ereditary V ertex D eletion . We believe that our recursive scheme can be applied to obtain O(log O(1) n )-approximation algorithms for many other problems as well.

Discrete simulations of genetic regulatory networks have been used to study subsystems of yeast successfully. Existing models underling these simulations are based on specific transition functions, which determine the node states in the network. However, implementations of existing models are not freely available and support only a textual user interface (TUI). Additionally, even if an implementation that supports a graphic user interface (GUI) were available, the computations necessary to analyze the simulations would still have to be done manually. Furthermore, the usage of different transition functions by existing models suggests that an enriched model is needed. We developed a software tool, called GRegNetSim, that allows the end-user (a biologist) to describe genetic regulatory networks graphically. The input is displayed visually via Cytoscape (an open-source platform for the representation and analysis of biological networks). The user can specify various transition functions at different nodes of the network, supporting, for example, threshold and gradient effects, thereby analyzing the network under a variety of modes dictated by these functions. GRegNetSim displays the relationship between the inputs and the mode of behavior of the network in a graphic form that is easy to interpret. Furthermore, it automatically extracts statistical data such as the percentage of simulations that reached a steady state or the percentage of simulations that terminated with a certain state. The discrete simulations performed by GRegNetSim can be used to elucidate and predict the behavior, structure, and properties of genetic regulatory networks in a unified manner. GRegNetSim is implemented as a Cytoscape App. Installation files, examples, and source code, along with a detailed user guide, are freely available at https://sites.google.com/site/gregnetsim/.

We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L subseteq Lines(P)={l: l is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L'subseteq L such that the graph G=(P,L') is isomorphic to a graph in F and L' has at most k crossings. By G=(P,L'), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L'. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build/draw/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s,t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links/roads (such as highways) may be cheaper to build and faster to traverse, and signals/moving objects would collide/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces. As a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d=n - then, P is in general position. The case of d=2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied - specifically, it is Crossing Minimization where G=(P,L) is a (bipartite) graph with a so called two-layer drawing. For d=2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k^2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k^2)-vertex kernel. Lastly, for graphs that contain an (s,t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP.

A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)^km epsilon^{-2})-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 +/- epsilon. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)^{k+O(log^3k)}m log n whenever epsilon^{-1}=k^{O(1)}. Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4^km epsilon^{-2})-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results. - We present a deterministic 4^{k+O(sqrt{k}(log^2k+log^2 epsilon^{-1}))}m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. - Additionally, we present a randomized 4^{k+O(log k(log k + log epsilon^{-1}))}m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method. Thus, the algorithm by Brand et al. runs in time 4^{k+o(k)}m whenever epsilon^{-1}=2^{o(k)}, while our deterministic and randomized algorithms run in time 4^{k+o(k)}m log n whenever epsilon^{-1}=2^{o(k^{1/4})} and epsilon^{-1}=2^{o(k/(log k))}, respectively. Prior to our work, no 2^{O(k)}n^{O(1)}-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth.

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs. Informally, our lemma states the following. For any map graph $G$, there exists a collection $(U_1,\ldots,U_t)$ of cliques of $G$ with the following property: $G$ either contains a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or it admits a tree decomposition where every bag is the union of $O(\sqrt{k})$ of the cliques in the above collection. The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the Connected Planar $\cal F$-Deletion problem (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions. For Longest Cycle/Path, these are the first subexponential-time parameterized algorithms on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known $2^{O({k^{0.75}\log{k}})} \cdot n^{O(1)}$-time algorithms on map graphs.

Single-elimination tournaments are a popular format in competitive environments. The Tournament Fixing Problem (TFP), which is the problem of finding a seeding of the players such that a certain player wins the resulting tournament, is known to be NP-hard in general and fixed-parameter tractable when parameterized by the feedback arc set number of the input tournament (an oriented complete graph) of expected wins/loses. However, the existence of polynomial kernelizations (efficient preprocessing) for TFP has remained open. In this paper, we present the first polynomial kernelization for TFP parameterized by the feedback arc set number of the input tournament. We achieve this by providing a polynomial-time routine that computes a SAT encoding where the number of clauses is bounded polynomially in the feedback arc set number.

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the linear programming dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT and ATT are fixed-parameter tractable, they can be solved in 2^{O(k log k)} n^{O(1)} time and 2^{O(k)} n^{O(1)} time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved in 2^{o(sqrt{k})} n^{O(1)} time under the Exponential-Time Hypothesis.

Given a graph G and an integer k, the Interval Vertex Deletion (IVD) problem asks whether there exists a subset S ⊆ V(G) of size at most k such that G–S is an interval graph. This problem is known to be NP-complete [Yannakakis, STOC’78]. Originally in 2012, Cao and Marx showed that IVD is fixed parameter tractable: they exhibited an algorithm with running time 10knO(1) [Cao and Marx, SODA’14]. The existence of a polynomial kernel for IVD remained a well-known open problem in Parameterized Complexity. In this paper, we settle this problem in the affirmative. We also introduce a “bounded intersection” variant of the classical Two Families theorem of Bollobás. We believe this result will find further applications in combinatorics and algorithm design.

An input to the Popular Matching problem, in the roommates setting, consists of a graph G where each vertex ranks its neighbors in strict order, known as its preference. In the Popular Matching problem the objective is to test whether there exists a matching M* such that there is no matching M where more people (vertices) are happier (in terms of the preferences) with M than with M*. In this paper we settle the computational complexity of the Popular Matching problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive integers r, k, decide whether G has an r-simple k-path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k. In the (r, k)-Monomial Detection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4(k/r)log r ·nO(1) for both problems. Gabizon et al. (2015) designed deterministic 2O((k/r)log r) · nO(1)-time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be noncanceling). Gabizon et al. also studied the following problem. In the p-Set (r, q)-Packing problem, given a universe V, positive integers p, q, r, and a collection ℋ of sets of size p whose elements belong to V, decide whether there exists a subcollection ℋ' of ℋ of size q where each element occurs in at most r sets of ℋ'. Gabizon et al. obtained a deterministic 2O((pq/r)log r) ·nO(1)-time algorithm for p-Set (r, q)-Packing. The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by the product of two parameters, that is, k/r and log r, where k = pq for p-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2o((k/r) log r) · (n + log k)O(1)-time algorithm for (r, k)-Monomial Detection, i.e. (r, k)-Monomial Detection is highly unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and p-Set (r, q)-Packing. We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e. whether there exists a computable function f such that the problems admit a f(k/r)(n + log k)O(1)-time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon zon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2O((k/r)2 log(k/r)) · (n + log k)O(1)-time algorithm for r-Simple k-Peth on digraphs and a 2O(k/r) ·(n+log k)O(1)-time algorithm for r-Simple k-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that p-Set (r, q)-Packing is FPT (in contrast, we prove that p-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detrction is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is noncanceling. For the special case of (r, k)-Monomial Detection here k is polynomially bounded by the input size (which is in XP), we show W[1]-hardness. Along the way to solve p-Set (r, q)-Packing, we obtain a polynomial kernel for any fixed p, which resolves a question posed by Gabizon et al. regarding the existence of polynomial kernels for problems with relaxed disjointness constraints. All our algorithms are deterministic.

We give a new decomposition theorem in unit disk graphs (UDGs) and demonstrate its applicability in the fields of Structural Graph Theory and Parameterized Complexity. First, our new decomposition theorem shows that the class of UDGs admits a Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010] and H-minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. Even bounded-degree UDGs can contain arbitrarily large cliques as minors, therefore our result is a significant advance in the study of contraction decompositions. Additionally, this result answers an open question posed by Hajiaghayi (www.youtube.com/watch?v=2Bq2gy1N01w) regarding the existence of contraction decompositions for classes of graphs beyond H-minor free graphs. Second, we present a “parameteric version” of our new decomposition theorem. We prove that there is an algorithm that given a UDG G and a positive integer k, runs in polynomial time and outputs a collection of O(k) tree decompositions of G with the following properties. Each bag in any of these tree decompositions can be partitioned into O(k) connected pieces (we call this measure the chunkiness of the tree decomposition). Moreover, for any subset S of at most k edges in G, there is a tree decomposition in the collection such that S is well preserved in the decomposition in the following sense. For any bag in the tree decomposition and any edge in S with both endpoints in the bag, either its endpoints lie in different pieces or they lie in a piece which is a clique. Having this decomposition at hand, we show that the design of parameterized algorithms for some cut problems becomes elementary. In particular, our algorithmic applications include single-exponential (or slightly superexponential) algorithms for well-studied problems such as Min Bisection, Steiner Cut, s-Way Cut, and Edge Multiway Cut-Uncut on UDGs; these algorithms are substantially faster than the best known algorithms for these problems on general graphs.

We give algorithms with running time $$2^{\mathcal {O}({\sqrt{k}\log {k}})} \cdot n^{\mathcal {O}(1)}$$ for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time $$2^{\mathcal {O}(k^{0.75}\log {k})} \cdot n^{\mathcal {O}(1)}$$ . Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to $$k^{\mathcal {O}(1)}$$ and there exists a solution that crosses every separator at most $$\mathcal {O}(\sqrt{k})$$ times. The running times of our algorithms are optimal up to the $$\log {k}$$ factor in the exponent, assuming the exponential time hypothesis.

The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdös--Pósa theorem in 1965, this problem received significant attention in the fields of graph theory and algorithm design. In particular, this problem is one of the first problems studied in the framework of parameterized complexity. The nonuniform fixed-parameter tractability of Cycle Packing follows from the Robertson--Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time $2^{\mathcal{O}(k^2)}\cdot |V|$ using exponential space. In the case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a $2^{\mathcal{O}(k\log^2k)}\cdot |V|$-time (deterministic) algorithm using exponential space, which is a consequence of the Erdös--Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on parameterized complexity. Yet, no algorithm that runs in time $2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, has been found. In light of this, it seems natural to ask whetherthe $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a $2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|$-time (deterministic) algorithm for Cycle Packing. In the case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, our algorithm runs in time linear in $|V|$, and its space complexity is polynomial in the input size.

The family of judicious partitioning problems, introduced by Bollobás and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection, and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is “judicious” in the sense that neither side is burdened by too many edges, and Balanced JB (BJB) also requires that the sizes of the sides themselves are “balanced” in the sense that neither of them is too large. Both of these problems were defined in the work by Bollobás and Scott and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is fixed parameter tractable (FPT) (which also proves that JB is FPT).

The question of the existence of a polynomial kernelization of the Vertex Cover Above LP problem was a long-standing, notorious open problem in parameterized complexity. Some years ago, the breakthrough work by Kratsch and Wahlström on representative sets finally answered this question in the affirmative [FOCS 2012]. In this paper, we present an alternative, algebraic compression of the Vertex Cover Above LP problem into the Rank Vertex Cover problem. Here, the input consists of a graph $G$, a parameter $k$, and a bijection between $V(G)$ and the set of columns of a representation of a matroid $M$, and the objective is to find a vertex cover whose rank is upper bounded by $k$.

Given a graph G and a parameter k , the C hordal V ertex D eletion (CVD) problem asks whether there exists a subset U ⊆ V ( G ) of size at most k that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size O ( k 161 log 58 k ) and asked whether one can design a kernel of size O ( k 10 ) [Jansen an Pilipczuk, SODA 2017]. While we do not completely resolve this question, we design a significantly smaller kernel of size O ( k 12 log 10 k ), inspired by the O ( k 2 ) -size kernel for F eedback V ertex S et [Thomassé, TALG 2010]. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.

M AX -C UT , E DGE D OMINATING S ET , G RAPH C OLORING , and H AMILTONIAN C YCLE on graphs of bounded clique-width have received significant attention as they can be formulated in MSO 2 (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO 1 (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g ( k ) n f ( k ) on graphs of clique-width k . Fomin et al. (2010) showed that the running times cannot be improved to g ( k ) n O (1) assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent f ( k ) in the running times. In a follow-up paper, Fomin et al. (2014) improved the running times for E DGE D OMINATING S ET and M AX -C UT to n O ( k ) , and proved that these problems cannot be solved in time g ( k ) n o ( k ) unless ETH fails. Thus, prior to this work, E DGE D OMINATING S ET and M AX -C UT were known to have tight n Θ ( k ) algorithmic upper and lower bounds. In this article, we provide lower bounds for H AMILTONIAN C YCLE and G RAPH C OLORING . For H AMILTONIAN C YCLE , our lower bound g ( k ) n o ( k ) matches asymptotically the recent upper bound n O ( k ) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight n Θ( k ) bounds for E DGE D OMINATING S ET , M AX -C UT , and H AMILTONIAN C YCLE , the G RAPH C OLORING problem has an upper bound of n O (2 k ) and a lower bound of merely n o (√ [4] k ) (implicit from the W[1]-hardness proof). In this article, we close the gap for G RAPH C OLORING by proving a lower bound of n 2 o ( k ) . This shows that G RAPH C OLORING behaves qualitatively different from the other three problems. To the best of our knowledge, G RAPH C OLORING is the first natural problem known to require exponential dependence on the parameter in the exponent of n .

We consider four well-studied NP-complete packing/covering problems on graphs: F eedback V ertex S et in T ournaments (FVST), C luster V ertex D eletion (CVD), T riangle P acking in T ournaments (TPT) and I nduced P 3 -P acking . For these four problems, kernels with O ( k 2 ) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-S et P acking ) or a hitting set of size at most k for a family of sets of size at most 3 (3-H itting S et ). In this article, we give the first kernels for FVST, CVD, TPT, and I nduced P 3 -P acking with a subquadratic number of vertices. Specifically, we obtain the following results. • FVST admits a kernel with O ( k 3/2 ) vertices. • CVD admits a kernel with O ( k 5/3 ) vertices. • TPT admits a kernel with O ( k 3/2 ) vertices. • I nduced P 3 -P acking admits a kernel with O ( k 5/3 ) vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with O( k 2−ϵ ) vertices for FVST and CVD. All of our results are based on novel uses of old and new “expansion lemmas” and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all ), (ii) almost none of the crown is used by the solution (as opposed to none ), and (iii) if H is removed from G , then there is almost no interaction between the head and the rest (as opposed to no interaction at all).

For a family of graphs $\cal F$, the canonical Weighted $\cal F$ Vertex Deletion problem is defined as follows: given an $n$-vertex undirected graph $G$ and a weight function $w: V(G)\rightarrow\mathbb{R}$, find a minimum weight subset $S\subseteq V(G)$ such that $G-S$ belongs to $\cal F$. We devise a recursive scheme to obtain $O(\log^{O(1)}n)$-approximation algorithms for such problems, building upon the classic technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to problems where an optimum solution $S$, together with a well-structured set $X$, form a balanced separator of $G$. We obtain the first $O(\log^{O(1)}n)$-approximation algorithms for the following problems. * We give an $O(\log^2n)$-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. * We give an $O(\log^3n)$-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth 1. Our methods also allow us to obtain in a clean fashion a $O(\log^{1.5}n)$-approximation algorithm for the Weighted $\cal F$ Vertex Deletion problem when $\cal F$ is a minor closed family excluding at least one planar graph. For the unweighted version of the problem constant factor approximation algorithms are were known~[Fomin et al., FOCS~2012], while for the weighted version considered here an $O(\log n \log\log n)$-approximation algorithm follows from~[Bansal et al., SODA~2017]. We believe that our recursive scheme can be applied to obtain $O(\log^{O(1)}n)$-approximation algorithms for many other problems as well.

Fradkin and Seymour [Journal of Combinatorial Graph Theory, Series B, 2015] defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms (a trait uncommon to parameterized algorithms) on tournaments, including Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk [ESA, 2013], where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is that the yes-instances of the problems above have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, an inductive argument and structural properties of the digraphs.

In the Directed Feedback Vertex Set (DFVS) problem, we are given as input a directed graph D and an integer k, and the objective is to check whether there exists a set S of at most k vertices such that F=D-S is a directed acyclic graph (DAG). Determining whether DFVS admits a polynomial kernel (parameterized by the solution size) is one of the most important open problems in parameterized complexity. In this article, we give a polynomial kernel for DFVS parameterized by the solution size plus the size of any treewidth-eta modulator, for any positive integer eta. We also give a polynomial kernel for the problem, which we call Vertex Deletion to treewidth-eta DAG, where given as input a directed graph D and a positive integer k, the objective is to decide whether there exists a set of at most k vertices, say S, such that D-S is a DAG and the treewidth of D-S is at most eta.

Given a Counting Monadic Second Order (CMSO) sentence $ψ$, the CMSO$[ψ]$ problem is defined as follows. The input to CMSO$[ψ]$ is a graph $G$, and the objective is to determine whether $G\models ψ$. Our main theorem states that for every CMSO sentence $ψ$, if CMSO$[ψ]$ is solvable in polynomial time on "globally highly connected graphs", then CMSO$[ψ]$ is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph $G$ and the task is to find a connected induced subgraph of $G$ such that "few" vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property.

A knockout tournament is a standard format of competition, ubiquitous in sports, elections and decision making. Such a competition consists of several rounds. In each round, all players that have not yet been eliminated are paired up into matches. Losers are eliminated, and winners are raised to the next round, until only one winner exists. Given that we can correctly predict the outcome of each potential match (modelled by a tournament D), a seeding of the tournament deterministically determines its winner. Having a favorite player v in mind, the Tournament Fixing Problem (TFP) asks whether there exists a seeding that makes v the winner. Aziz et al. [AAAI’14] showed that TFP is NP-hard. They initiated the study of the parameterized complexity of TFP with respect to the feedback arc set number k of D, and gave an XP-algorithm (which is highly inefficient). Recently, Ramanujan and Szeider [AAAI’17] showed that TFP admits an FPT algorithm, running in time 2^{ O(k^2 log k)} n ^{O(1)}. At the heart of this algorithm is a translation of TFP into an algebraic system of equations, solved in a black box fashion (by an ILP solver). We present a fresh, purely combinatorial greedy solution. We rely on new insights into TFP itself, which also results in the better running time bound of 2^{ O(k log k)} n^{ O(1)} . While our analysis is intricate, the algorithm itself is surprisingly simple.

In a tournament, $n$ players enter the competition. In each round, they are paired-up to compete against each other. Losers are thrown, while winners proceed to the next round, until only one player (the winner) is left. Given a prediction of the outcome, for every pair of players, of a match between them (modeled by a digraph $D$), the competitive nature of a tournament makes it attractive for manipulators. In the Tournament Fixing (TF) problem, the goal is to decide if we can conduct the competition (by controlling how players are paired-up) so that our favorite player $w$ wins. A common form of manipulation is to bribe players to alter the outcome of matches. Kim and Williams [IJCAI 2015] integrated such deceit into TF, and showed that the resulting problem is NP-hard when $\ell<(1-\epsilon)\log n$ alterations are possible (for any fixed $\epsilon>0$). For this problem, our contribution is fourfold. First, we present two operations that ``obfuscate deceit'': given one solution, they produce another solution. Second, we present a combinatorial result, stating that there is always a solution with all reversals incident to $w$ and ``elite players''. Third, we give a closed formula for the case where $D$ is a DAG. Finally, we present exact exponential-time and parameterized algorithms for the general case.

Deterministic polynomial-time computation of a representation of a transversal matroid is a longstanding open problem. We present a deterministic computation of a so-called union representation of a transversal matroid in time quasipolynomial in the rank of the matroid. More precisely, we output a collection of linear matroids such that a set is independent in the transversal matroid if and only if it is independent in at least one of them. Our proof directly implies that if one is interested in preserving independent sets of size at most r, for a given r\in\mathbb{N}, but does not care whether larger independent sets are preserved, then a union representation can be computed deterministically in time quasipolynomial in r. This consequence is of independent interest, and sheds light on the power of union~representation. Our main result also has applications in Parameterized Complexity. First, it yields a fast computation of representative sets, and due to our relaxation in the context of r, this computation also extends to (standard) truncations. In turn, this computation enables to efficiently solve various problems, such as subcases of subgraph isomorphism, motif search and packing problems, in the presence of color lists. Such problems have been studied to model scenarios where pairs of elements to be matched may not be identical but only similar, and color lists aim to describe the set of compatible elements associated with each element.

Hubs are high-degree nodes within a network. The examination of the emergence and centrality of hubs lies at the heart of many studies of complex networks such as telecommunication networks, biological networks, social networks and semantic networks. Furthermore, identifying and allocating hubs are routine tasks in applications. In this paper, we do not seek a hub that is a single node, but a hub that consists of k nodes. Formally, given a graph G=(V,E), we a seek a set A subseteq V of size k that induces a connected subgraph from which at least p edges emanate. Thus, we identify k nodes which can act as a unit (due to the connectivity constraint) that is a hub (due to the cut constraint). This problem, which we call Multi-Node Hub (MNH), can also be viewed as a variant of the classic Max Cut problem. While it is easy to see that MNH is W[1]-hard with respect to the parameter k, our main contribution is the first parameterized algorithm that shows that MNH is FPT with respect to the parameter p. Despite recent breakthrough advances for cut-problems like Multicut and Minimum Bisection, MNH is still very challenging. Not only does a connectivity constraint has to be handled on top of the involved machinery developed for these problems, but also the fact that MNH is a maximization problem seems to prevent the applicability of this machinery in the first place. To deal with the latter issue, we give non-trivial reduction rules that show how MNH can be preprocessed into a problem where it is necessary to delete a bounded-in-parameter number of vertices. Then, to handle the connectivity constraint, we use a novel application of the form of tree decomposition introduced by Cygan et al. [STOC 2014] to solve Minimum Bisection, where we demonstrate how connectivity constraints can be replaced by simpler size constraints. Our approach may be relevant to the design of algorithms for other cut-problems of this nature.

In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph G and an integer ruυ for every pair of vertices u, υ ∊ V(G). The objective is to construct a subgraph H of minimum weight which contains ruυ edge-disjoint (or node-disjoint) u-υ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. Consequently, there is a long line of research into exact-polynomial time algorithms as well as approximation algorithms for various restrictions of this problem. An important restriction of this problem is one where the connectivity demands are the same for every pair of vertices. In this paper, we first consider the edge-connectivity version of this problem which we call λ-Edge Connected Subgraph (λ-ECS). In this problem, the input is a λ-edge connected (di)graph G and an integer k and the objective is to check whether G contains a spanning subgraph H that is also λ-edge connected and H excludes at least k edges of G. In other words, we are asked to compute a maximum subset of edges, of cardinality at least k, which may be safely deleted from G without affecting its connectivity. If we replace λ-edge connectivity with λ-vertex connectivity we get the λ-Vertex Connected Subgraph (λ-VCS) problem. We show that λ-ECS is fixed-parameter tractable (FPT) for both graphs and digraphs even if the (di)graph has nonnegative real weights on the edges and the objective is to exclude from H, some edges of G whose total weight exceeds a prescribed value. In particular, we design an algorithm for the weighted variant of the problem with running time 2O(k log k) |V(G)|O(1). We follow up on this result and obtain a polynomial compression for λ-ECS on unweighted graphs. As a direct consequence of our results, we obtain the first FPT algorithm for the parameterized version of the classical Minimum Equivalent Graph (MEG) problem. We also show that λ-Ves is FPT on digraphs; however the problem on undirected graphs remains open. Finally, we complement our algorithmic findings by showing that SNDP is W[1]-hard for both arc and vertex connectivity versions on digraphs. The core of our algorithms is composed of new combinatorial results on connectivity in digraphs and undirected graphs.

Max-Cut (MC), Edge Dominating Set (EDS), Graph Coloring (GC) and Hamiltonian Path (HP) on graphs of bounded cliquewidth have received significant attention as they can be formulated in MSO2 (and therefore have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle's theorem), but cannot be formulated in MSO1 (which would have yielded linear-time algorithms on bounded cliquewidth graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g(k)nf(k) on graphs of cliquewidth k. Fomin et al. [Intractability of Clique-Width Parameterizations. SIAM J. Comput. 39(5): 1941–1956 (2010)] showed that the running times cannot be improved to g(k)nO(1) assuming W[1]≠FPT. However, this does not rule out nontrivial improvements to the exponent f(k) in the running times. In a follow-up paper, Fomin et al. [Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width. SIAM J. Comput. 43(5): 1541–1563 (2014)] improved the running times for EDS and MC to nO(k), and proved g(k)no(k) lower bounds for EDS, MC and HP assuming the ETH. Recently, Bergougnoux, Kante and Kwon [WADS 2017] gave an nO(k)-time algorithm for HP. Thus, prior to this work, EDS, MC and HP were known to have tight nΘ(k) algorithmic upper and lower bounds. In contrast, GC has an upper bound of nO(2k) and a lower bound of merely (implicit from the W[1]-hardness proof). In this paper, we close the gap for GC by proving a lower bound of n2o(k) This shows that GC behaves qualitatively different from the other three problems. To the best of our knowledge, GC is the first natural problem known to require exponential dependence on the parameter in the exponent of n.

We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P3-Packing. For these four problems kernels with O(k2) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this paper, we give the first kernels for FVST, CVD, TPT and Induced P3-Packing with a subquadratic number of vertices. Specifically, we obtain the following results. • FVST admits a kernel with vertices. • CVD admits a kernel with vertices. • TPT admits a kernel with vertices. • Induced P3-Packing admits a kernel with vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a d-degenerate graph G and an integer k, outputs an independent set Y, such that for every independent set X in G of size at most k, the probability that X is a subset of Y is at least . The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = {{s1, t1}, {s2, t2}, …, {sℓ, tℓ}} of terminal pairs and an integer k, returns an induced subgraph G* of G that maintains all the inclusion minimal multicuts of G of size at most k, and does not contain any (k + 2)-vertex connected set of size 2O(k). In particular, G* excludes a clique of size 2O(k) as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable s-t Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on d-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.

A classical result by Erdos and Posa states that there is a function $f: {\mathbb N} \rightarrow {\mathbb N}$ such that for every $k$, every graph $G$ contains $k$ pairwise vertex disjoint cycles or a set $T$ of at most $f(k)$ vertices such that $G-T$ is acyclic. The generalisation of this result to directed graphs is known as Younger's conjecture and was proved by Reed, Robertson, Seymour and Thomas in 1996. This so-called Erdos-Posa-property can naturally be generalised to arbitrary graphs and digraphs. Robertson and Seymour proved that a graph $H$ has the Erdos-Posa-property if, and only if, $H$ is planar. In this paper we study the corresponding problem for digraphs. We obtain a complete characterisation of the class of strongly connected digraphs which have the Erdos-Posa-property (both for topological and butterfly minors). We also generalise this result to classes of digraphs which are not strongly connected. In particular, we study the class of vertex-cyclic digraphs (digraphs without trivial strong components). For this natural class of digraphs we obtain a nearly complete characterisation of the digraphs within this class with the Erdos-Posa-property. In particular we give positive and algorithmic examples of digraphs with the Erdos-Posa-property by using directed tree decompositions in a novel way.

For a target rank $r$, the rigidity of a matrix $A$ over a field $\mathbb{F}$ is the minimum Hamming distance between $A$ and a matrix of rank at most $r$. Rigidity is a classical concept in computational complexity theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of parameterized complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been neglected. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of parameterized complexity. Naturally, given parameters $r$ and $k$, the Matrix Rigidity problem asks whether the rigidity of $A$ for the target rank $r$ is at most $k$. We show that in the case $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}$ is any finite field, this problem is fixed-parameter tractable with respect to $k+r$. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in real algebraic geometry, which are not well known in parameterized complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a \sf W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of Matrix Rigidity, highlighting the different flavors of this problem.

Given a 1.5-dimensional terrain T , also known as an x -monotone polygonal chain, the T errain G uarding problem seeks a set of points of minimum size on T that guards all of the points on T . Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T . The T errain G uarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm (SODA 2005). However, only in 2010 King and Krohn (SODA 2010) finally showed that T errain G uarding is NP-hard. In spite of the remarkable developments in approximation algorithms for T errain G uarding , next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether, if parameterized by the size k of a solution guard set, it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this article, we answer the first question affirmatively by developing an n O (√ k ) -time algorithm for both D iscrete T errain G uarding and C ontinuous T errain G uarding . We also make non-trivial progress with respect to the second question: we show that D iscrete O rthogonal T errain G uarding , a well-studied special case of T errain G uarding , is fixed-parameter tractable.

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.

In the Maximum-Duo Preservation String Mapping (Max-Duo PSM) problem, the input consists of two related strings A and B of length n and a nonnegative integer k. The objective is to determine whether there exists a mapping m from the set of positions of A to the set of positions of B that maps only to positions with the same character and preserves at least k duos, which are pairs of adjacent positions. We develop a randomized algorithm that solves Max-Duo PSM in time 4^k * n^{O(1)}, and a deterministic algorithm that solves this problem in time 6.855^k * n^{O(1)}. The previous best known (deterministic) algorithm for this problem has running time (8e)^{2k+o(k)} * n^{O(1)} [Beretta et al., Theor. Comput. Sci. 2016]. We also show that Max-Duo PSM admits a problem kernel of size O(k^3), improving upon the previous best known problem kernel of size O(k^6).

The family of judicious partitioning problems, introduced by Bollob\'as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\'as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).

The Cycle Packing problem asks whether a given undirected graph G=(V,E) contains k vertex-disjoint cycles. Since the publication of the classic Erdos-Posa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson–Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time 2^{O(k^2)}|V| using exponential space. In case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a 2^{O(k\log^2 k)}|V|-time (deterministic) algorithm using exponential space, which is a consequence of the Erdos-Posa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time 2^{o(k\log^2k)}|V|^{O(1)}, beating the bound 2^{O(k\log^2k)}\cdot |V|^{O(1)}, has been found. In light of this, it seems natural to ask whether the 2^{O(k\log^2k)}|V|^{O(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a 2^{O(k\log^2k/log log k})} |V|-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the known bound, our algorithm runs in time linear in |V|, and its space complexity is polynomial in the input size.

Given a graph G and a parameter k, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset U ⊆ V (G) of size at most k that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size O(k161 log58 k), and asked whether one can design a kernel of size O(k10). While we do not completely resolve this question, we design a significantly smaller kernel of size O(k25 log14 k), inspired by the O(k2)-size kernel for Feedback Vertex Set. To obtain this result, we first design an O(opt-log2 n)-factor approximation algorithm for CVD, which is central to our kernelization procedure. Thus, we improve upon both the kernelization algorithm and the approximation algorithm of Jansen and Pilipczuk. Next, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution. We believe that this notion could be useful in designing kernels for other problems.

The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this paper, we examine an important family of graphs, namely the family of split graphs, which in the context of edge contractions, is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k, the Split Contraction problem asks whether there exists a subset X of edges of G such that G/X is a split graph and X has at most k elements. Here, G/X is the graph obtained from G by contracting edges in X. It was previously claimed that the Split Contraction problem is fixed-parameter tractable. However, we show that, despite its deceptive simplicity, it is W[1]-hard. Our main result establishes the following conditional lower bound: under the Exponential Time Hypothesis, the Split Contraction problem cannot be solved in time 2^(o(l^2)) * poly(n) where l is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2^(o(l^2)) * poly(n) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.

The rigidity of a matrix A for a target rank r over a field F is the minimum Hamming distance between A and a matrix of rank at most r. Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of Parameterized Complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been neglected. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Naturally, given parameters r and k, the Matrix Rigidity problem asks whether the rigidity of A for the target rank r is at most k. We show that in case F equals the reals or F is any finite field, this problem is fixed-parameter tractable with respect to k+r. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in Real Algebraic Geometry, which are not well known in Parameterized Complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of Matrix Rigidity, highlighting the different flavors of this problem.

We show two results related to finding trees and paths in graphs. First, we show that in $O^*(1.657^k2^{l/2})$ time one can either find a $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph or conclude that such a tree does not exist. Our solution can be applied as a subroutine to solve the $k$-Internal Spanning Tree problem in $O^*(min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the running time can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for Hamiltonicity and $k$-Path in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8. Our results extend the technique by Björklund [SIAM J. Comput., 43 (2014), pp. 280--299] and Björklund et al. [Narrow Sieves for Parameterized Paths and Packings, CoRR, arXiv:1007. 1161, 2010] to finding structures more general than paths as well as refine it to handle special classes of graphs more efficiently.

The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest studying the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximum Minimal Vertex Cover (MMVC) problem, which has applications to wireless ad hoc networks and whose minimization version, Vertex Cover, is one of the most studied problems in the field of parameterized complexity. We first present tight conditional lower bounds for the running time of any algorithm for MMVC or its weighted variant. Next, we develop a parameterized approximation algorithm for MMVC and its weighted variant. The approximation ratio of this algorithm cannot be achieved by polynomial-time algorithms unless P = NP, and its running time cannot be matched by exact parameterized algorithms unless the strong exponential time hypothesis fails. In particular, the algorithm defines a user-controlled parameter that corresponds to a trade-off between time and approximation ratio.

Problems of genome rearrangement are central in both evolution and cancer. Most evolutionary scenarios have been studied under the assumption that the genome contains a single copy of each gene. In contrast, tumor genomes undergo deletions and duplications, and thus the number of copies of genes varies. The number of copies of each gene along a chromosome is called its copy number profile. Understanding copy number profile changes can assist in predicting disease progression and treatment. To date, questions related to distances between copy number profiles gained little scientific attention. Here we focus on the following fundamental problem, introduced by Schwarz et al. (PLOS Comp. Biol., 2014): given two copy number profiles, u and v, compute the edit distance from u to v, where the edit operations are segmental deletions and amplifications. We establish the computational complexity of this problem, showing that it is solvable in linear time and constant space.

The classic K-Cycle problem asks if a graph G, with vertex set V(G), has a simple cycle containing all vertices of a given set K subseteq V(G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K subset of V(G) and an injective coloring c from K to {1,2,...,|K|}, decide if G has a simple cycle containing each color in {1,2,...,|K|} (once). Another problem widely known since the introduction of color coding is {Colorful Cycle}. Given a graph G and a coloring c from V(G) to {1,2,...,k} for some natural number k, it asks if G has a simple cycle of length k containing each color in {1,2,...,k} (once). We study a generalization of these problems: Given a graph G, a set K subset of V(G), a list-coloring L from K to 2^{{1,2,...,k^*}} for some natural number k^* and a parameter k, List K-Cycle asks if one can assign a color to each vertex in K so that G would have a simple cycle (of arbitrary length) containing exactly k vertices from K with distinct colors. We design a randomized algorithm for List K-Cycle running in time 2^kn^{O(1)} on an -vertex graph, matching the best known running times of algorithms for both K-Cycle and Colorful Cycle. Moreover, unless the Set Cover Conjecture is false, our algorithm is essentially optimal. We also study a variant of List K-Cycle that generalizes the classic Hamiltonicity problem, where one specifies the size of a solution. Our results integrate three related algebraic approaches, introduced by Bjorklund, Husfeldt and Taslaman (SODA'12), Bjorklund, Kaski and Kowalik (STACS'13), and Bjorklund (FOCS'10).

In this paper we consider Simultaneous Feedback Edge Set (Sim-FES) problem. In this problem, the input is an $n$-vertex graph $G$, an integer $k$ and a coloring function ${\sf col}: E(G) \rightarrow 2^{[α]}$ and the objective is to check whether there is an edge subset $S$ of cardinality at most $k$ in $G$ such that for all $i \in [α]$, $G_i - S$ is acyclic. Here, $G_i=(V(G), \{e\in E(G) \mid i \in {\sf col}(e)\})$ and $[α]=\{1,\ldots,α\}$. When $α=1$, the problem is polynomial time solvable. We show that for $α=3$ Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic graphs. The same reduction shows that the problem does not admit an algorithm of running time $O(2^{o(k)}n^{O(1)})$ unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time $O(2^{ωkα+α\log k} n^{O(1)})$, where $ω$ is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when $α=2$. We also give a kernel for Sim-FES with $(kα)^{O(α)}$ vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph $G$, an integer $q$ and, a coloring function ${\sf col}: E(G) \rightarrow 2^{[α]}$. The question is whether there is a edge subset $F$ of cardinality at least $q$ in $G$ such that for all $i\in [α]$, $G[F_i]$ is acyclic. Here, $F_i=\{e \in F \mid i \in \textsf{col}(e)\}$. We give an FPT algorithm for running in time $O(2^{ωq α}n^{O(1)})$.

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D - S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k^3) to O(k^2) and of Pumpkin Vertex Deletion Set from O(k^18) to O(k^3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions.

In this work, we study the well-known $r$-Dimensional $k$-Matching ($(r,k)$-DM), and $r$-Set $k$-Packing ($(r,k)$-SP) problems. Given a universe $U := U_1 \uplus \cdots \uplus U_r$ and an $r$-uniform family $\mathcal{F} \subseteq U_1 \times \cdots \times U_r$, the $(r,k)$-DM problem asks if $\mathcal{F}$ admits a collection of $k$ mutually disjoint sets. Given a universe $U$ and an $r$-uniform family $\mathcal{F}\subseteq 2^U$, the $(r,k)$-SP problem asks if $\mathcal{F}$ admits a collection of $k$ mutually disjoint sets. We employ techniques based on dynamic programming and representative families. This leads to a deterministic algorithm with running time $\mathcal{O} (2.851^{(r-1)k}\cdot|\mathcal{F}|\cdot n\log^2 n\cdot \log W)$ for the weighted version of $(r,k)$-DM, where $W$ is the maximum weight in the input, and a deterministic algorithm with running time $\mathcal{O}(2.851^{(r-0.5501)k}\cdot|\mathcal{F}|\cdot n\log^2 n\cdot \log W)$ for the weighted version of $(r,k)$-SP. Thus, we significantly improve the previous best known deterministic running times for $(r,k)$-DM and $(r,k)$-SP and the previous best known running times for their weighted versions. We rely on structural properties of $(r,k)$-DM and $(r,k)$-SP to develop algorithms that are faster than those that can be obtained by a standard use of representative sets. Incorporating the principles of iterative expansion, we obtain a better algorithm for $(3,k)$-DM, running in time $\mathcal{O}(2.004^{3k}\cdot|\mathcal{F}| \cdot n\log^2 n)$. We believe that this algorithm demonstrates an interesting application of representative families in conjunction with more traditional techniques. Furthermore, we present kernels of size $\mathcal{O}(e^rr(k-1)^r\log W)$ for the weighted versions of $(r,k)$-DM and $(r,k)$-SP, improving the previous best known kernels of size $\mathcal{O}(r!r(k-1)^r\log W)$ for these problems.