We study a Stackelberg variant of the classical Most Vital Links problem, modeled as a one-round adversarial game between an attacker and a defender. The attacker strategically removes up to k edges from a flow network to maximally disrupt flow between a source s and a sink t, after which the defender optimally reroutes the remaining flow. To capture this attacker–defender interaction, we introduce a new mathematical model of discounted cuts, in which the cost of a cut is evaluated by excluding its k most expensive edges. This model generalizes the Most Vital Links problem and uncovers novel algorithmic and complexity-theoretic properties. We develop a unified algorithmic framework for analyzing various forms of discounted cut problems, including minimizing or maximizing the cost of a cut under discount mechanisms that exclude either the k most expensive or the k cheapest edges. While most variants are NP-complete on general graphs, our main result establishes polynomial-time solvability for all discounted cut problems in our framework when the input is restricted to bounded-genus graphs, a relevant class that includes many real-world networks such as transportation and infrastructure networks. With this work, we aim to open collaborative bridges between artificial intelligence, algorithmic game theory, and operations research.

In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: 1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on 𝒪(1) rows (for fixed values of the parameters). 2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. 3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.

We study the problem of finding a family of diverse minimum edge s-t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s-t cuts C_1, …, C_k such that for any i,j ∈ [k], the number of edges in the symmetric difference C_i △ C_j is at least d. For d ∈ {1,2}, the problem corresponds to counting minimum s-t cuts in G, which is #P-complete [Provan and Ball, SICOMP 1983]. The problem is also known to be NP-complete already for k = 3 [de Berg, López Martínez, Spieksma, ISAAC 2024]. Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d. The main ingredients of our FPT algorithm build on novel structural properties of diverse minimum s-t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlström [JACM 2025].

Rank aggregation seeks a representative permutation for a collection of rankings and plays a central role in areas such as social choice, information retrieval, and computational biology. Two fundamental aggregation tasks are the center and median problems, which minimize the maximum and the total distance to the input permutations, respectively. While these problems are well understood under Kendall’s tau and related distances, their parameterized complexity under the Ulam metric, an edit-distance-based metric on permutations, has remained largely unexplored. In this work, we initiate a systematic study of the parameterized complexity of rank aggregation under the Ulam metric. We consider both the center and median problems, as well as their generalizations to the k-center and k-median clustering settings, parameterized by the number of centers k and the distance budget d (corresponding to the maximum distance for center variants and the total distance for median variants). Both problems are known to be NP-hard already for k = 1. We show that the Ulam k-center problem remains NP-hard when d = 1, but is fixed-parameter tractable when parameterized by k + d. Our algorithm is based on a novel local-search framework tailored to the non-local nature of Ulam distances. We complement this by proving that no polynomial kernel exists for the k+d parameterization unless NP ⊆ coNP/poly. For the Ulam k-median problem parameterized by the total distance d, we establish W[1]-hardness and provide an XP algorithm. We also provide a polynomial kernel for the parameter k + d, which in turn yields a fixed-parameter tractable algorithm.

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 \(2^{\mathcal O(\sqrt{k}\log k)} + n \cdot k^{\mathcal O(1)}\).

We introduce a series of graph decompositions based on the modulator/target scheme of modification problems that enable several algorithmic applications that parametrically extend the algorithmic potential of planarity. In the core of our approach is a polynomial time algorithm for computing planar \(\mathcal{H}\)-modulators. Given a graph class \(\mathcal{H}\), a planar \(\mathcal{H}\)-modulator of a graph \(G\) is a set \(X \subseteq V(G)\) such that the “torso” of \(X\) is planar and all connected components of \(G-X\) belong to \(\mathcal{H}\). Here, the torso of \(X\) is obtained from \(G[X]\) if, for every connected component of \(G-X\), we form a clique out of its neighborhood on \(G[X]\). We introduce \(\mathcal{H}\)-Planarity as the problem of deciding whether a graph \(G\) has a planar \(\mathcal{H}\)-modulator. We prove that, if \(\mathcal{H}\) is hereditary, CMS0-definable, and decidable in polynomial time, then \(\mathcal{H}\)-Planarity is solvable in polynomial time.

The question of "what can be computed locally?" lies at the heart of distributed computing in networks. As established in Naor and Stockmeyer's seminal paper (STOC 1993, Edsger W. Dijkstra Prize in Distributed Computing 2025), this question is undecidable, even for graph problems whose solutions can be checked locally. In this paper, we adopt a novel perspective on the question, by asking for which classes Π of problems, and for which classes G of graphs, all problems in Π can be solved efficiently in a distributed manner in all graphs of G. This paper focuses on two natural candidates for such an approach, namely the class of problems expressible in first-order logic (FO), because they possess an intrinsic form of locality thanks to Gaifman's theorem, and the class of graphs with bounded expansion, because they form a large class of graphs encompassing, e.g., planar, bounded-genus, bounded-treewidth, and bounded-degree graphs, as well as graphs excluding a fixed minor or topological minor, sparse Erdös--Rényi graphs (a.a.s.), and several network models such as stochastic block models for suitable parameter ranges.

The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth’s theorem on posets and Kőnig’s theorem on matchings in bipartite graphs. The theorem asserts that for every graph G, the vertex set of G can be partitioned into at most α(G) vertex-disjoint paths, where α(G) is the maximum size of an independent set in G. The proof of the Gallai-Milgram theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most α(G) vertex-disjoint paths. While the Gallai-Milgram theorem is tight—there are graphs where one really needs α(G) paths, not fewer, to cover the vertex set of G—it was not known prior to our work whether deciding if a graph G could be covered by fewer than α(G) vertex-disjoint paths can be done in polynomial time. We resolve this question by proving the following algorithmic extension of the Gallai–Milgram theorem for undirected graphs: There is an algorithm that, for an n-vertex graph G and an integer parameter k ≥ 1, runs in time 22O(k4logk) · nO(1) and outputs a path cover P of G together with either a correct conclusion that P is a minimum-size path cover or an independent set of size |P| + k, certifying that P contains at most α(G) − k paths. Thus, for k ∈ O((loglogn)1/4−ε) our algorithm runs in polynomial time, and either computes a minimum-size path cover of G, or finds a path cover of size at most α(G) − k. We find the existence of such an algorithm quite surprising for the following reason. The problems of computing a path cover and a maximum independent set are both notoriously hard, yet our algorithm either solves one of them or provides meaningful information about the other. The proof of our algorithmic extension of the Gallai–Milgram theorem is non-trivial and builds on several novel algorithmic ideas. One of the key subroutines in our algorithm is an FPT algorithm, parameterized by α(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest—prior to our work, no polynomial-time algorithm for deciding Hamiltonicity was known, even for graphs with independence number at most three. Moreover, the algorithmic techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment. We show that all these problems are FPT when parameterized by the independence number of the graph. Notably, the independence-number parameterization departs from the typical direction of research in parameterized complexity. First, α(G) measures a graph’s density, whereas most prior work in the area focuses on parameters describing sparsity, such as treewidth or vertex cover. Second, most structural parameters studied in parameterized complexity can be computed exactly or well-approximated in polynomial or even FPT time, whereas computing α(G) is notoriously difficult from almost any computational perspective. The fact that it can nevertheless serve as the basis for efficient parameterization is particularly striking.

Abstract We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths . In this problem, the input is an edge-weighted (directed or undirected) n -vertex graph G along with k terminal pairs $$(s_1,t_1),(s_2,t_2),\ldots ,(s_k,t_k)$$ ( s 1 , t 1 ) , ( s 2 , t 2 ) , … , ( s k , t k ) . The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA ’21], which demonstrates the polynomial-time solvability of the problem for a fixed value of k . Lochet’s result implies the existence of a polynomial-time ck -approximation for Maximum Vertex-Disjoint Shortest Paths , where $$c \le 1$$ c ≤ 1 is a constant. (One can guess 1/ c terminal pairs to connect in $$k^{O({1}/{c})}$$ k O ( 1 / c ) time and then utilize Lochet’s algorithm to compute the solution in $$n^{f({1}/{c})}$$ n f ( 1 / c ) time.) Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an o ( k )-approximation within $$f(k){{\,\textrm{poly}\,}}(n)$$ f ( k ) poly ( n ) time for any function f that only depends on k . Our second result demonstrates the infeasibility of achieving an approximation ratio of $$m^{{1}/{2}-\varepsilon }$$ m 1 / 2 - ε in polynomial time, unless P $$=$$ = NP. We also show that this bound is tight by providing a simple $$\sqrt{\ell }$$ ℓ -approximation algorithm, where $$\ell $$ ℓ

Abstract We study the following Independent Stable Set problem. Let $$\varvec{G}$$ G be an undirected graph and $$\varvec{\mathcal {M}} \varvec{=} \varvec{(V(G),} \varvec{\mathcal {I})}$$ M = ( V ( G ) , I ) be a matroid whose elements are the vertices of $$\varvec{G}$$ G . For an integer $$\varvec{k}\varvec{\ge } \varvec{1}$$ k ≥ 1 , the task is to decide whether $$\varvec{G}$$ G contains a set $$\varvec{S}\varvec{\subseteq } \varvec{V(G)}$$ S ⊆ V ( G ) of size at least $$\varvec{k}$$ k which is independent (stable) in $$\varvec{G}$$ G and independent in $$\varvec{\mathcal {M}}$$ M . This problem generalizes several well-studied algorithmic problems, including Rainbow Independent Set , Rainbow Matching , and Bipartite Matching with Separation . We show that when the matroid $$\varvec{\mathcal {M}}$$ M is represented by an independence oracle, then for any computable function $$\varvec{f}$$ f , no algorithm can solve Independent Stable Set using $$\varvec{f(k)} \varvec{\cdot } \varvec{n}^{\varvec{o(k)}}$$ f ( k ) · n o ( k ) calls to the oracle. On the other hand, when the graph $$\varvec{G}$$ G

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund and Husfeldt, ICALP 2014; Mari et al., SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing . In Min-Sum Cycle Packing , we try to find, in a weighted undirected graph, \( k \) vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed \( k \) , solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by \( k \) . The same results hold for the version of the problem where the task is to find \( k \) edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing , which is a special case of Min-Sum Cycle Packing that asks to find a packing of \( k \) shortest cycles in a graph. We prove this problem to be Fixed-Parameter Tractable (FPT) when parameterized by \( k \) on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Whether Min-Sum Cycle Packing is FPT on planar graphs, or Shortest Cycle Packing on general graphs, remains open.

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP -hard graph problems, like maximum-weight independent set, can be solved in time \(n^{\mathcal{O}(k)}\) if the input \( n \) -vertex graph is given together with a tree decomposition of independence number \( k \) . Yolov, in SODA 2018, gave an algorithm that, given an \( n \) -vertex graph \( G \) and an integer \( k \) , in time \(n^{\mathcal{O}(k^{3})}\) either constructs a tree decomposition of \( G \) whose independence number is \(\mathcal{O}(k^{3})\) or correctly reports that the tree-independence number of \( G \) is larger than \( k \) . In this article, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time \(2^{\mathcal{O}(k^{2})}n^{\mathcal{O}(k)}\) and either outputs a tree decomposition of \( G \) with independence number at most \(8k\) or determines that the tree-independence number of \( G \) is larger than \( k \) . This implies \(2^{\mathcal{O}(k^{2})}n^{\mathcal{O}(k)}\) -time algorithms for various problems, like maximum-weight independent set, parameterized by the tree-independence number \( k \) without needing the decomposition as an input. Assuming Gap-ETH, an \(n^{\Omega(k)}\) factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP -hard: We show that for every constant \(k\geq 4\) it is NP -complete to decide whether a given graph has the tree-independence number at most \( k \) .

According to the classic Chvátal’s Lemma from 1977, a graph \( G \) of minimum degree \(\delta(G)\) contains every tree on \(\delta(G)+1\) vertices. Our main result is the following algorithmic “extension” of Chvátal’s Lemma: For any \( n \) -vertex graph \( G \) , an integer \( k \) , and a tree \( T \) on at most \(\delta(G)+k\) vertices, deciding whether \( G \) contains a subgraph isomorphic to \( T \) can be done in time \(f(k)\cdot n^{\mathcal{O}(1)}\) for some function \( f \) of \( k \) only. The proof is based on an intricate interplay between extremal graph theory and parameterized algorithms.

In the problem Fault-Tolerant Path ( FTP ), we are given an edge-weighted directed graph G = ( V , E ) , a subset U ⊆ E of vulnerable edges, two vertices s, t ∈ V , and integers k and ℓ. The task is to decide whether there exists a subgraph H of G with total cost at most ℓ such that, after the removal of any k vulnerable edges, H still contains an s − t -path. We study whether Fault-Tolerant Path is fixed-parameter tractable (FPT) and whether it admits a polynomial kernel under various parameterizations. Our choices of parameters include: the number of vulnerable edges in the input graph, the number of safe (i.e, invulnerable) edges in the input graph, the budget ℓ, the minimum number of safe edges in any optimal solution, the minimum number of vulnerable edges in any optimal solution, the required redundancy k , and natural above- and below-guarantee parameterizations. We provide an almost complete description of the complexity landscape of FTP for these parameters.

Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in ℝ^d (d ≥ 1), or equivalently, distance spaces that can be isometrically embedded in ℝ^d. In this work, we investigate whether a distance space can be isometrically embedded in ℝ^d after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing, asks whether an input distance space on n points can be transformed, using at most k modifications, into a space that is isometrically embeddable in ℝ^d. We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves Euclidean Embedding Editing in time (dk)^𝒪(d+k) + n^𝒪(1). The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of Euclidean Embedding Editing with 𝒪((dk)²) points. For the special but important case of Euclidean Embedding Editing where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of min{(d+3)^k, 2^{d+k}} ⋅ n^𝒪(1). Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most 2 ⋅ Opt outliers in time 2^d ⋅ n^{𝒪(1)}. This 2-approximation algorithm improves upon the previous (3+ε)-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations.

We investigate the problem of constructing fault-tolerant bases in matroids. Given a matroid M and a redundancy parameter k, a k-fault-tolerant basis is a minimum-size set of elements such that, even after the removal of any k elements, the remaining subset still spans the entire ground set. Since matroids generalize linear independence across structures such as vector spaces, graphs, and set systems, this problem unifies and extends several fault-tolerant concepts appearing in prior research. Our main contribution is a fixed-parameter tractable (FPT) algorithm for the k-fault-tolerant basis problem, parameterized by both k and the rank r of the matroid. This two-variable parameterization by k + r is shown to be tight in the following sense. On the one hand, the problem is already NP-hard for k=1. On the other hand, it is Para-NP-hard for r \geq 3 and polynomial-time solvable for r \leq 2.

Covering and partitioning the edges of a graph into cliques are classical problems at the intersection of combinatorial optimization and graph theory, having been studied through a range of algorithmic and complexity-theoretic lenses. Despite the well-known fixed-parameter tractability of these problems when parameterized by the total number of cliques, such a parameterization often fails to be meaningful for sparse graphs. In many real-world instances, on the other hand, the minimum number of cliques in an edge cover or partition can be very close to the size of a maximum independent set α(G). Motivated by this observation, we investigate above αparameterizations of the edge clique cover and partition problems. Concretely, we introduce and study Edge Clique Cover Above Independent Set (ECC/α) and Edge Clique Partition Above Independent Set (ECP/α), where the goal is to cover or partition all edges of a graph using at most α(G) + k cliques, and k is the parameter. Our main results reveal a distinct complexity landscape for the two variants. We show that ECP/αis fixed-parameter tractable, whereas ECC/αis NP-complete for all k \geq 2, yet can be solved in polynomial time for k \in {0,1}. These findings highlight intriguing differences between the two problems when viewed through the lens of parameterization above a natural lower bound. Finally, we demonstrate that ECC/αbecomes fixed-parameter tractable when parameterized by k + ω(G), where ω(G) is the size of a maximum clique of the graph G. This result is particularly relevant for sparse graphs, in which ωis typically small. For H-minor free graphs, we design a subexponential algorithm of running time f(H)^{\sqrt{k}}n^{O(1)}.

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.

We establish that for every first-order logic (FO) formula ϕ, which captures a vast number of computational problems on graphs, and every graph class G of bounded expansion, there exists a deterministic distributed algorithm that, for any n-node graph G ∈ G with diameter D, determines whether G ⊨ ϕ within O(D+log n) rounds in the standard CONGEST model. Graph classes of bounded expansion encompass many well-known families of sparse graphs, including planar graphs, bounded-genus graphs, bounded-treedepth graphs, bounded-treewidth graphs, bounded-degree graphs, graphs that exclude a fixed graph H as a minor or topological minor, random graphs with constant average degree (a.a.s.), and many network models (e.g., stochastic block models) for some ranges of parameters.

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014;

In the Hedge Cut problem, the edges of a graph are partitioned into groups called hedges, and the question is what is the minimum number of hedges to delete to disconnect the graph. Ghaffari, Karger, and Panigrahi [SODA 2017] showed that Hedge Cut can be solved in quasipolynomial-time, raising the hope for a polynomial time algorithm. Jaffke, Lima, Masarik, Pilipczuk, and Souza [SODA 2023] complemented this result by showing that assuming the Exponential Time Hypothesis (ETH), no polynomial-time algorithm exists. In this paper, we show that Hedge Cut is fixed-parameter tractable parameterized by the solution size ℓ by providing an algorithm with running time , which can be upper bounded by Cℓ · (n + m)O (1) for any constant c > 1. This running time captures at the same time the fact that the problem is quasipolynomial-time solvable, and that it is fixed-parameter tractable parameterized by ℓ. We further generalize this algorithm to an algorithm with running time for Hedge k-CUT.

The irrelevant vertex technique provides a powerful tool for the design of parameterized algorithms for a wide variety of problems on graphs. A common characteristic of these problems, permitting the application of this technique on surface-embedded graphs, is the fact that every graph of large enough treewidth contains a vertex that is irrelevant, in the sense that its removal yields an equivalent instance of the problem. The straightforward application of this technique yields algorithms with running time that is quadratic in the size of the input graph. This running time is due to the fact that it takes linear time to detect one irrelevant vertex and the total number of irrelevant vertices to be detected is linear as well. Using advanced techniques, sub-quadratic algorithms have been designed for particular problems, even in general graphs. However, designing a general framework for linear-time algorithms has been open, even for the bounded-genus case. In this paper we introduce a general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus. Our technique consists in decomposing any surface-embedded graph into a tree-structured collection of bounded-treewidth subgraphs where detecting globally irrelevant vertices can be done locally and independently. Our method is applicable to a wide variety of known graph containment or graph modification problems where the irrelevant vertex technique applies. Examples include the (Induced) Minor Folio problem, the (Induced) Disjoint Paths problem, and the F-Minor-Deletion problem.

Let G be a weighted graph embedded in a metric space (M, d_M). The vertices of G correspond to the points in M, with the weight of each edge uv being the distance d_M(u,v) between their respective points in M. The dilation (or stretch) of G is defined as the minimum factor t such that, for any pair of vertices u,v, the distance between u and v - represented by the weight of a shortest u,v-path - is at most t⋅ d_M(u,v). We study Dilation t-Augmentation, where the objective is, given a metric M, a graph G, and numerical values k and t, to determine whether G can be transformed into a graph with dilation t by adding at most k edges. Our primary focus is on the scenario where the metric M is the shortest path metric of an unweighted graph Γ. Even in this specific case, Dilation t-Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by k when Γ is a complete graph, already for t = 2. Our main contribution lies in providing new insights into the impact of combinations of various parameters on the computational complexity of the problem. We establish the following. - The parameterized dichotomy of the problem with respect to dilation t, when the graph G is sparse: Parameterized by k, the problem is FPT for graphs excluding a biclique K_{d,d} as a subgraph for t ≤ 2 and the problem is W[1]-hard for t ≥ 3 even if G is a forest consisting of disjoint stars. - The problem is FPT parameterized by the combined parameter k+t+Δ, where Δ is the maximum degree of the graph G or Γ.

This paper considers four problems with possible applications in network design: Given a graph G with | G | = n and an integer k ≥ 0 , does G have a DFS tree with (i) ≤ k leaves, (ii) ≥ k leaves, (iii) ≤ n − k leaves, and (iv) ≥ n − k leaves? We show that all four problems are NP-hard. When parameterized by k , we prove that while (i) is para-NP-hard and (ii) is W[1]-hard, both (iii) and (iv) admit polynomial kernels with O ( k 3 ) vertices, implying FPT algorithms running in k O ( k ) ⋅ n O ( 1 ) time. Our polynomial kernels are based on a O ( k ) -sized vertex cover structure associated with the solution of these problems. As a byproduct, we obtain polynomial kernels for these problems parameterized by the vertex cover number of the input graph.

Abstract. A framework consists of an undirected graph [Formula: see text] and a matroid [Formula: see text] whose elements correspond to the vertices of [Formula: see text]. Recently, Fomin et al. [ Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2023, pp. 2214–2227] and Eiben, Koana, and Wahlström [ Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2024, pp. 377–423] developed parameterized algorithms for computing paths of rank [Formula: see text] in frameworks. More precisely, for vertices [Formula: see text] and [Formula: see text] of [Formula: see text], and an integer [Formula: see text], they gave FPT algorithms parameterized by [Formula: see text] deciding whether there is an [Formula: see text]-path in [Formula: see text] whose vertex set contains a subset of elements of [Formula: see text] of rank [Formula: see text]. These algorithms are based on the Schwartz–Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph [Formula: see text] is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid [Formula: see text] is represented over rationals. We complement this result by proving that if the input matroids are given by their independence oracles, then there is no algorithm solving the problem with [Formula: see text] oracle queries. Furthermore, this computational lower bound holds even if the input graphs are planar graphs of treewidth at most two.

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence , defining some property of the modified part of the graph, and the target sentence , defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic ( CMSO ) and have models of bounded treewidth, while target sentences express first-order logic ( FO ) properties. Our logic captures problems that are not definable in FO and, moreover, may have instances of unbounded treewidth. Our main result is that, for this compound logic, model-checking can be done in quadratic time on minor-free graphs. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique . It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle’s theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle’s theorem and Gaifman’s theorem. Our algorithmic meta-theorem encompasses, unifies, and extends the known meta-algorithmic results for CMSO and FO on minor-closed graph classes.

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.

The present bias is a well-documented behavioral trait that significantly influences human decision-making, with present-biased agents often prioritizing immediate rewards over long-term benefits, leading to suboptimal outcomes in various real-world scenarios. Kleinberg and Oren (2014) proposed a popular graph-theoretical model of inconsistent planning to capture the behavior of present-biased agents. In this model, a multi-step project is represented by a weighted directed acyclic task graph, where the agent traverses the graph based on present-biased preferences. We use the model of Kleinberg and Oren to address the principal-agent problem, where a principal, fully aware of the agent’s present bias, aims to modify an existing project by adding or deleting tasks. The challenge is to create a modified project that satisfies two somewhat contradictory conditions. On one hand, the present-biased agent should select specific tasks deemed important by the principal. On the other hand, if the anticipated costs in the modified project become too high for the agent, there is a risk of the agent abandoning the entire project, which is not in the principal’s interest. To tackle this issue, we leverage the tools of parameterized complexity to investigate whether the principal’s strategy can be efficiently identified. We provide algorithms and complexity bounds for this problem.

Vertex (s, t)-Cut and Vertex Multiway Cut are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation R ∈ 𝔽^{r × n} of a linear matroid ℳ = (V(G), ℐ) of rank r in the input, and the goal is to determine whether there exists a vertex subset S ⊆ V(G) that has the required cut properties, as well as is independent in the matroid ℳ. We refer to these problems as Independent Vertex (s, t){-cut}, and Independent Multiway Cut, respectively. We show that these problems are fixed-parameter tractable (FPT) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid ℳ). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al. STOC '22], combined with a dynamic programming algorithm on flow-paths á la [Feige and Mahdian, STOC '06] that maintains a representative family of solutions w.r.t. the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain FPT algorithms for the independent version of Odd Cycle Transversal. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph $G$ and two sets of vertices $S$ and $T$. The question is, what is the minimum number of edges to remove from $G$, such that we separate all of $S$ from all of $T$, while maintaining that every vertex in $S$, and respectively in $T$, stays in the same connected component. We show that this problem can be solved in time $2^{|S|+|T|} n^{O(1)}$ with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number $r$ of faces in the plane graph covering the terminals $S \cup T$, by providing an algorithm of running time $4^{r + O(\sqrt r)} n^{O(1)}$.

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^{O(k)} if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov, in [SODA 2018], gave an algorithm that, given an n-vertex graph G and an integer k, in time n^{O(k^3)} either constructs a tree decomposition of G whose independence number is O(k^3) or correctly reports that the tree-independence number of G is larger than k. In this paper, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time 2^{O(k^2)} n^{O(k)} and either outputs a tree decomposition of G with independence number at most $8k$, or determines that the tree-independence number of G is larger than k. This implies 2^{O(k^2)} n^{O(k)}-time algorithms for various problems, like maximum weight independent set, parameterized by the tree-independence number k without needing the decomposition as an input. Assuming Gap-ETH, an n^{Ω(k)} factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k \ge 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition.

According to the classic Chvátal's Lemma from 1977, a graph of minimum degree δ(G) contains every tree on δ(G) + 1 vertices. Our main result is the following algorithmic “extension” of Chvátal's Lemma: For any n-vertex graph G, integer k, and a tree T on at most δ(G) + k vertices, deciding whether G contains a subgraph isomorphic to T, can be done in time f (k) · nO(1) for some function f of k only.

Abstract We initiate the study of theDiverse Pair of (Maximum/ Perfect) Matchingsproblems which given a graphGand an integerk, ask whetherGhas two (maximum/perfect) matchings whose symmetric difference is at leastk.Diverse Pair of Matchings(asking for two not necessarily maximum or perfect matchings) is $$\textsf{NP}$$ NP -complete on general graphs ifkis part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show thatDiverse Pair of Maximum Matchingsis $$\textsf{FPT}$$ FPT parameterized byk. We round off the work by showing thatDiverse Pair of Matchingshas a kernel on $${\mathcal {O}}(k^2)$$ O(k2) vertices.

Abstract Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit “natural” guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than $$m/2 +k$$ m / 2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least $$n/2 +k$$ n / 2 + k for some $$k\ge 1$$ k ≥ 1 and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree $$\delta (G)\le n/2$$ δ ( G ) ≤ n / 2 , the length of a longest cycle L is at least $$2\delta (G)$$ 2 δ ( G ) . Thus the “essential” part in finding the longest cycle is in approximating the “offset” $$k = L - 2 \delta (G)$$ k = L - 2 δ ( G ) . The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length $$2\delta (G)+\Omega (f(k))$$ 2 δ ( G ) + Ω ( f ( k ) ) .

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 study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0≤α≤1, the question is whether there is a set S⊆V of size k with a specified coverage function covα(S) at least p (or at most p for the minimization version). The coverage function covα(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1−α. α-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n−k)-Cut. A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greedy vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α>0 and subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α>1/3 and minimization with α<1/3.4

Abstract. In 1959, Erdős and Gallai proved that every graph [Formula: see text] with average vertex degree [Formula: see text] contains a cycle of length at least [Formula: see text]. We provide an algorithm that for [Formula: see text], in time [Formula: see text] decides whether a 2-connected [Formula: see text]-vertex graph [Formula: see text] contains a cycle of length at least [Formula: see text]. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function f defined on the edges (or the vertices) of an undirected graph G , we seek for a cycle C in G of minimum cost 𝖮𝖯𝖳 = f(C) . We give an algorithm that given an n -vertex graph G , parameter ɛ > 0, and the function f represented by an oracle, in time n 𝒪 (log 1/ɛ) finds a cycle C in G with f(C) ≤ (1+ɛ). 𝖮𝖯𝖳. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest ( s,t -Path problem, which requires exponentially many queries to the oracle for finding an n 2/3-ɛ -approximation Goel et al. [ 7 ], FOCS 2009. We complement our algorithm with a matching lower bound. We show that for every ɛ > 0, obtaining a (1+ɛ)-approximation requires at least n Ω (log 1/ ɛ) queries to the oracle. When the function f is integer-valued, our algorithm yields that a cycle of cost 𝖮𝖯𝖳 can be found in time n 𝒪(log 𝖮𝖯𝖳) . In particular, for 𝖮𝖯𝖳 = n 𝒪(1) this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that n 𝒪(log n ) queries are required even when 𝖮𝖯𝖳= 𝒪( n ). We also consider special cases of monotone submodular functions, corresponding to the number of different color classes needed to cover a cycle in an edge-colored multigraph G . For special cases of the corresponding minimization problem, we obtain fixed-parameter tractable algorithms and polynomial-time algorithms, when restricted to certain classes of inputs.

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored \(n\) -vertex undirected graph, vertices \(s\) and \(t\) , and an integer \(k\) , finds an \((s,t)\) -path containing at least \(k\) different colors in time \(2^{k}n^{\mathcal{O}(1)}\) . This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman on finding a path through \(k\) specified vertices. It also implies the first \(2^{k}n^{\mathcal{O}(1)}\) time algorithm for finding an \((s,t)\) -path of length at least \(k\) . Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \(n\) -vertex undirected graph \(G\) , a matroid \(M\) whose elements correspond to the vertices of \(G\) and which is represented over a finite field of order \(q\) , a positive integer weight function on the vertices of \(G\) , two sets of vertices \(S,T\subseteq V(G)\) , and integers \(p,k,w\) , and the task is to find \(p\) vertex-disjoint paths from \(S\) to \(T\) so that the union of the vertices of these paths contains an independent set of \(M\) of cardinality \(k\) and weight \(w\) , while minimizing the sum of the lengths of the paths. We give a \(2^{p+\mathcal{O}(k^{2}\log(q+k))}n^{\mathcal{O}(1)}w\) time randomized algorithm for this problem.

The task of the broadcast problem is, given a graph G and a source vertex s, to compute the minimum number of rounds required to disseminate a piece of information from s to all vertices in the graph. It is assumed that, at each round, an informed vertex can transmit the information to at most one of its neighbors. The broadcast problem is known to NP-hard. We show that the problem is FPT when parametrized by the size k of a feedback edge set, or by the size k of a vertex cover, or by k=n−t, where t is the input deadline for the broadcast protocol to complete.

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.

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit "natural" guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2 + k for some k ≥ 1 and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ(G) ≤ n/2, the length of the longest cycle L is at least 2δ(G). Thus the "essential" part of finding the longest cycle is in approximating the "offset" k = L - 2δ(G). The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k)).

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or $\mathcal{G}$-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem. We finally prove extensions where the target property is expressible in FOL+DP, i.e., the enhancement of FOL with disjoint-paths predicates.

A framework consists of an undirected graph $G$ and a matroid $M$ whose elements correspond to the vertices of $G$. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank $k$ in frameworks. More precisely, for vertices $s$ and $t$ of $G$, and an integer $k$, they gave FPT algorithms parameterized by $k$ deciding whether there is an $(s,t)$-path in $G$ whose vertex set contains a subset of elements of $M$ of rank $k$. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph $G$ is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid $M$ is represented over rationals. Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

We study the kernelization of exploration problems on temporal graphs. A temporal graph consists of a finite sequence of snapshot graphs $\mathcal{G}=(G_1, G_2, \dots, G_L)$ that share a common vertex set but might have different edge sets. The non-strict temporal exploration problem (NS-TEXP for short) introduced by Erlebach and Spooner, asks if a single agent can visit all vertices of a given temporal graph where the edges traversed by the agent are present in non-strict monotonous time steps, i.e., the agent can move along the edges of a snapshot graph with infinite speed. The exploration must at the latest be completed in the last snapshot graph. The optimization variant of this problem is the $k$-arb NS-TEXP problem, where the agent's task is to visit at least $k$ vertices of the temporal graph. We show that under standard computational complexity assumptions, neither of the problems NS-TEXP nor $k$-arb NS-TEXP allow for polynomial kernels in the standard parameters: number of vertices $n$, lifetime $L$, number of vertices to visit $k$, and maximal number of connected components per time step $γ$; as well as in the combined parameters $L+k$, $L + γ$, and $k+γ$. On the way to establishing these lower bounds, we answer a couple of questions left open by Erlebach and Spooner. We also initiate the study of structural kernelization by identifying a new parameter of a temporal graph $p(\mathcal{G}) = \sum_{i=1}^{L} (|E(G_i)|) - |V(G)| +1$. Informally, this parameter measures how dynamic the temporal graph is. Our main algorithmic result is the construction of a polynomial (in $p(\mathcal{G})$) kernel for the more general Weighted $k$-arb NS-TEXP problem, where weights are assigned to the vertices and the task is to find a temporal walk of weight at least $k$.

We introduce the following submodular generalization of the SHORTEST CYCLE problem. For a nonnegative monotone submodular cost function f defined on the edges (or the vertices) of an undirected graph G, we seek for a cycle C in G of minimum cost OPT = f(C). We give an algorithm that given an n-vertex graph G, parameter ε > 0, and the function f represented by an oracle, in time n

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is a randomized algorithm, that given a colored n-vertex undirected graph, vertices s and t, and an integer k, finds an (s,t)-path containing at least k different colors in time 2kn

The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate dpk(x1,y1,…, xk,yk), expressing the existence of internally vertex-disjoint paths between Xi and yi, for i ∈ {1,…, k}. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every proper minor-closed graph class, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP, namely the scattered disjoint paths logic, FOL+SDP, where we further consider the atomic predicate s-sdpk(x1,y1,…,xk,yk), demanding that the disjoint paths are within distance bigger than some fixed value s. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus. * The full version of the paper can be accessed at https://arxiv.org/abs/2211.01723

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependencies on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k-means and k-median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the “best explanation” by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

We introduce the rendezvous game with adversaries. In this game, two players, Facilitator and Divider, play against each other on a graph. Facilitator has two agents and Divider has a team of k agents located in some vertices. They take turns in moving their agents to adjacent vertices (or staying put). Facilitator wins if his agents meet in some vertex. Divider aims to prevent the rendezvous of Facilitator's agents. We show that deciding whether Facilitator can win is PSPACE-hard and, when parameterized by k, co-W[2]-hard. Moreover, even deciding whether Facilitator can win within τ steps is co-NP-complete already for τ=2. On the other hand, for chordal and P5-free graphs, we prove that the problem is solvable in polynomial time. Finally, we show that the problem is fixed-parameter tractable parameterized by both the graph's neighborhood diversity and the number of steps τ.

We study two “above guarantee” versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least distG(s,t)+k. Bezáková et al. [7] proved that on undirected graphs the problem is fixed-parameter tractable (FPT). Our first main result establishes a connection between Longest Detour on directed graphs and 3- Disjoint Paths on directed graphs. Using these new insights, we design a 2O(k)⋅nO(1) time algorithm for the problem on directed planar graphs. Furthermore, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k. We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs.

In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a1,…,an} over Σm, where Σ is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable for the binary case Σ={0,1}. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters are lower and upper bounded by integer parameters p and q, respectively. Our main theorem is that the problem is solvable in time 2O(Blog⁡B)|Σ|B⋅(mn)O(1).

Abstract In this work, we study the k-median clustering problem with an additional equal-size constraint on the clusters from the perspective of parameterized preprocessing. Our main result is the first lossy (2-approximate) polynomial kernel for this problem parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existence of an (exact) kernel of polynomial size and a PTAS.

Abstract. A linkage in a graph [Formula: see text] of size [Formula: see text] is a subgraph [Formula: see text] of [Formula: see text] whose connected components are [Formula: see text] paths. The pattern of a linkage of size [Formula: see text] is the set of [Formula: see text] pairs formed by the endpoints of these paths. A consequence of the Unique Linkage Theorem is the following: there exists a function [Formula: see text] such that if a plane graph [Formula: see text] contains a sequence [Formula: see text] of at least [Formula: see text] nested cycles and a linkage of size at most [Formula: see text] whose pattern vertices lay outside the outer cycle of [Formula: see text] then [Formula: see text] contains a linkage with the same pattern avoiding the inner cycle of [Formula: see text]. In this paper we prove the following variant of this result: Assume that all the cycles in [Formula: see text] are “orthogonally” traversed by a linkage [Formula: see text] and [Formula: see text] is a linkage whose pattern vertices may lay either outside the outer cycle or inside the inner cycle of [Formula: see text]. We prove that there are two functions [Formula: see text], such that if [Formula: see text] has size at most [Formula: see text], [Formula: see text] has size at least [Formula: see text] and [Formula: see text], then there is a linkage with the same pattern as [Formula: see text] that is “internally combed” by [Formula: see text], in the sense that [Formula: see text]. This result applies to any graph that is partially embedded on a disk (where [Formula: see text] is also embedded). In fact, we prove this result in the most general version where the linkage [Formula: see text] is [Formula: see text]-scattered: every two vertices of distinct paths are within a distance bigger than [Formula: see text]. We deduce several variants of this result in the cases where [Formula: see text] and [Formula: see text]. These variants permit the application of the Unique Linkage Theorem on several path routing problems on embedded graphs.

For a finite collection of graphs ℱ, the ℱ- TM-Deletion problem has as input an n -vertex graph G and an integer k and asks whether there exists a set S ⊆ V(G) with |S| ≤ k such that G \ S does not contain any of the graphs in ℱ as a topological minor. We prove that for every such ℱ, ℱ - TM-Deletion is fixed parameter tractable on planar graphs. Our algorithm runs in a 2 𝒪( k 2) ⋅ n 2 time, or, alternatively, in 2 𝒪( k ) ⋅ n 4 time. Our techniques can easily be extended to graphs that are embeddable on any fixed surface.

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k-means and k-median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the ``best explanation'' by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 0 in time 2^{O(k)} n^{O(1)} decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

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.

In this paper, we consider the Minimum-Load k-Clustering/Facility Location (MLkC) problem where we are given a set P of n points in a metric space that we have to cluster and an integer k > 0 that denotes the number of clusters. Additionally, we are given a set F of cluster centers in the same metric space. The goal is to select a set C ⊆ F of k centers and assign each point in P to a center in C, such that the maximum load over all centers is minimized. Here the load of a center is the sum of the distances between it and the points assigned to it. Although clustering/facility location problems have rich literature, the minimum-load objective has not been studied substantially, and hence MLkC has remained a poorly understood problem. More interestingly, the problem is notoriously hard even in some special cases including the one in line metrics as shown by Ahmadian et al. [APPROX 2014, ACM Trans. Algorithms 2018]. They also show APX-hardness of the problem in the plane. On the other hand, the best-known approximation factor for MLkC is O(k), even in the plane. In this work, we study a fair version of MLkC inspired by the work of Chierichetti et al. [NeurIPS, 2017]. Here the input points are partitioned into 𝓁 protected groups, and only clusters that proportionally represent each group are allowed. MLkC is the special case with 𝓁 = 1. For the fair version, we are able to obtain a randomized 3-approximation algorithm in f(k,𝓁)⋅ n^O(1) time. Also, our scheme leads to an improved (1 + ε)-approximation in the case of Euclidean norm with the same running time (depending also linearly on the dimension d). Our results imply the same approximations for MLkC with running time f(k)⋅ n^O(1), achieving the first constant-factor FPT approximations for this problem in general and Euclidean metric spaces.

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every n-vertex 2-connected graph G with minimum vertex degree δ ≥ 2 contains a cycle with at least min{2δ, n} vertices. In particular, if δ ≥ n/2, then G is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than 2δ + 1. Also, there are non-Hamiltonian graphs with all vertices but one of degree at least n/2. This prompts naturally to the following algorithmic questions. For k ≥ 1, (A) How difficult is to decide whether a 2-connected graph contains a cycle of length at least min{2δ + k, n}? (B) How difficult is to decide whether a graph G is Hamiltonian, when at least n–k vertices of G are of degrees at least n/2–k? The first question was asked by Fomin, Golovach, Lokshtanov, Panolan, Saurabh, and Zehavi. The second question is due to Jansen, Kozma, and Nederlof. Even for a very special case of k = 1, the existence of a polynomial-time algorithm deciding whether G contains a cycle of length at least min{2δ + 1, n} was open. We resolve both questions by proving the following algorithmic generalization of Dirac's theorem: If all but k vertices of a 2-connected graph G are of degree at least δ, then deciding whether G has a cycle of length at least min{2δ + k, n} can be done in time . The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own.

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bez\'akov\'a et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O (k)} n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path Above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path Above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path Above Diameter is solvable in polynomial time for each k\in{1,\dots, 4} and is NP-complete for every k\geq 5. The parameterized complexity of Longest Path Above Diameter on general directed graphs remains an interesting open problem.

Abstract We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

A subset T⊆V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every set of vertices of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.

An enumeration kernel as defined by Creignou et al. (2017) [11] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem’s fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph property P expressible by a first order-logic formula \( \varphi \in \Sigma _3 \) , that is, of the form \( \begin{equation*} \varphi =\exists x_1\exists x_2\cdots \exists x_r\ \ \forall y_{1}\forall y_{2}\cdots \forall y_{s}\ \ \exists z_1\exists z_2\cdots \exists z_t~~ \psi ,\end{equation*} \) where \( \psi \) is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed \( k \) , is fixed-parameter tractable parameterized by \( k \) . Properties of graphs expressible by formulas from \( \Sigma _3 \) include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas \( \varphi \in \Pi _3 \) , for which computing elimination distance is \( {\sf W}[2] \) -hard.

De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorithms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension $d \ge 2$ on intersection graphs of similarly-sized fat objects many well-known graph problems including Independent Set, $r$-Dominating Set for constant $r$, Cycle Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable in time $2^{O(n^{1-1/d})}$ and within polynomial space.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as $L(1,1)$-Labelling). A classical complexity result on Colouring is a well-known dichotomy for $H$-free graphs (a graph is $H$-free if it does not contain $H$ as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on $H$-free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours $k$ is fixed, that is, not part of the input. From our study it follows that for fixed $k$ the three problems behave in the same way, but this is no longer true if $k$ is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

The elimination distance to some target graph propertyPis a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph propertyPexpressible by a first order-logic formula φ ∈ Σ3, that is, of the form φ = ∃x1∃x2⋯∃xr∀y1∀y2⋯∀ys∃z1∃z2⋯∃ztψ, where ψ is a quantifier-free first-order formula, checking whether the elimination distance of a graph toPdoes not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Σ3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas φ ∈ Π3, for which computing elimination distance is W[2]-hard.

We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers l (the number of irrelevant features) and k (the number of clusters), budget B, and a set of n categorical data points (represented by m-dimensional vectors whose elements belong to a finite set of values Σ), we want to select m-l relevant features such that the cost of any optimal k-clustering on these features does not exceed B. Here the cost of a cluster is the sum of Hamming distances (l0-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters k, B, and |Σ|. Our main result is an algorithm that solves the Feature Selection problem in time f(k,B,|Σ|)⋅m^{g(k,|Σ|)}⋅n² for some functions f and g. In other words, the problem is fixed-parameter tractable parameterized by B when |Σ| and k are constants. Our algorithm for Feature Selection is based on a solution to a more general problem, Constrained Clustering with Outliers. In this problem, we want to delete a certain number of outliers such that the remaining points could be clustered around centers satisfying specific constraints. One interesting fact about Constrained Clustering with Outliers is that besides Feature Selection, it encompasses many other fundamental problems regarding categorical data such as Robust Clustering, Binary and Boolean Low-rank Matrix Approximation with Outliers, and Binary Robust Projective Clustering. Thus as a byproduct of our theorem, we obtain algorithms for all these problems. We also complement our algorithmic findings with complexity lower bounds.

We consider a generalization of the fundamental k-means clustering for data with incomplete or corrupted entries. When data objects are represented by points in ℝd, a data point is said to be incomplete when some of its entries are missing or unspecified. An incomplete data point with at most Δ unspecified entries corresponds to an axis-parallel affine subspace of dimension at most Δ, called a Δ-point. Thus we seek a partition of n input Δ-points into k clusters minimizing the k-means objective. For Δ = 0, when all coordinates of each point are specified, this is the usual k-means clustering. We give an algorithm that finds an (1 + ∊)-approximate solution in time f(k, ∊, Δ) · n2 · d for some function f of k, ∊, and Δ only.

We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem consists of a matroid M, a weight function ω:E(M)→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k bases B_1, ..., B_k of M such that the weight of the symmetric difference of any pair of these bases is at least d. This is a diverse variant of the classical matroid base packing problem. The input to the Weighted Diverse Common Independent Sets problem consists of two matroids M₁,M₂ defined on the same ground set E, a weight function ω:E→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k common independent sets I_1, ..., I_k of M₁ and M₂ such that the weight of the symmetric difference of any pair of these sets is at least d. This is motivated by the classical weighted matroid intersection problem. The input to the Diverse Perfect Matchings problem consists of a graph G and integers k ≥ 1, d ≥ 0. The task is to decide if G contains k perfect matchings M_1, ..., M_k such that the symmetric difference of any two of these matchings is at least d. The underlying problem of finding one solution (basis, common independent set, or perfect matching) is known to be doable in polynomial time for each of these problems, and Diverse Perfect Matchings is known to be NP-hard for k = 2. We show that Weighted Diverse Bases and Weighted Diverse Common Independent Sets are both NP-hard. We show also that Diverse Perfect Matchings cannot be solved in polynomial time (unless P=NP) even for the case d = 1. We derive fixed-parameter tractable (FPT) algorithms for all three problems with (k,d) as the parameter. The above results on matroids are derived under the assumption that the input matroids are given as independence oracles. For Weighted Diverse Bases we present a polynomial-time algorithm that takes a representation of the input matroid over a finite field and computes a poly(k,d)-sized kernel for the problem.

Abstract We study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L -

In k-Clustering we are given a multiset of n vectors X⊂Zd and a nonnegative number D, and we need to decide whether X can be partitioned into k clusters C1,…,Ck such that the cost∑i=1kminci∈Rd⁡∑x∈Ci‖x−ci‖pp≤D, where ‖⋅‖p is the Lp-norm. For p=2, k-Clustering is k-Means. We study k-Clustering from the perspective of parameterized complexity. The problem is known to be NP-hard for k=2 and also for d=2. It is a long-standing open question, whether the problem is fixed-parameter tractable (FPT) for the combined parameter d+k. In this paper, we focus on the parameterization by D. We complement the known negative results by showing that for p=0 and p=∞, k-Clustering is W[1]-hard when parameterized by D. Interestingly, we discover a tractability island of k-Clustering: for every p∈(0,1], k-Clustering is solvable in time 2O(Dlog⁡D)(nd)O(1).

We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choice of parameterization is strongly influenced by the work of Biró, Hujter, and Tuza, who in 1992 introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi-)graph $H$. In this work, we turn to proper $H$-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph $H$ is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results. Path Cover admits a kernel of size $\mathcal{O}(\|H\|^8)$, where $\|H\|$ is the size of graph $H$. In other words, we design an algorithm that for an $n$-vertex graph $G$ and integer $k\geq 1$, in time polynomial in $n$ and $\|H\|$, outputs a graph $G'$ of size $\mathcal{O}(\|H\|^8)$ and $k'\leq |V(G')|$ such that the vertex set of $G$ is coverable by $k$ vertex-disjoint paths if and only if the vertex set of $G'$ is coverable by $k'$ vertex-disjoint paths. Hamiltonian Cycle admits a kernel of size $\mathcal{O}(\|H\|^8)$. Cycle Cover admits a polynomial kernel. We prove it by providing a compression of size $\mathcal{O}(\|H\|^{10})$ into another \sf NP-complete problem, namely, Prize Collecting Cycle Cover, that is, we design an algorithm that, in time polynomial in $n$ and $\|H\|$, outputs an equivalent instance of Prize Collecting Cycle Cover of size $\mathcal{O}(\|H\|^{10})$. In all our algorithms we assume that a proper $H$-decomposition is given as a part of the input.

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs $G$ and $H$ are 2-isomorphic, or equivalently, their cycle matroids are isomorphic if and only if $G$ can be transformed into $H$ by a series of operations called Whitney switches. In this paper we consider the quantitative question arising from Whitney's theorem: Given two 2-isomorphic graphs, can we transform one into another by applying at most $k$ Whitney switches? This problem is already \sf NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size $\mathcal{O}(k)$ and thus is fixed-parameter tractable when parameterized by $k$.

We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show that for every ε ∈ (0, 1), there is a {randomized} (1+ε)-approximation algorithm for 𝓁₁-Rank-r Approximation over {GF}(2) of running time m^{O(1)}n^{O(2^{4r}⋅ ε^{-4})}. This is the first polynomial time approximation scheme (PTAS) for this problem.

The incidence matrix of a graph is a fundamental object naturally appearing in many applications, involving graphs such as social networks, communication networks, or transportation networks. Often, the data collected about the incidence relations can have some slight noise. In this paper, we initiate the study of the computational complexity of recovering incidence matrices of graphs from a binary matrix: given a binary matrix M which can be written as the superposition of two binary matrices L and S, where S is the incidence matrix of a graph from a specified graph class, and L is a matrix (i) of small rank or, (ii) of small (Hamming) weight. Further, identify all those graphs whose incidence matrices form part of such a superposition. Here, L represents the noise in the input matrix M. Another motivation for this problem comes from the Matroid Minors project of Geelen, Gerards and Whittle, where perturbed graphic and co-graphic matroids play a prominent role. There, it is expected that a perturbed binary matroid (or its dual) is presented as L+S where L is a low rank matrix and S is the incidence matrix of a graph. Here, we address the complexity of constructing such a decomposition. When L is of small rank, we show that the problem is NP-complete, but it can be decided in time (mn)^O(r), where m,n are dimensions of M and r is an upper-bound on the rank of L. When L is of small weight, then the problem is solvable in polynomial time (mn)^O(1). Furthermore, in many applications it is desirable to have the list of all possible solutions for further analysis. We show that our algorithms naturally extend to enumeration algorithms for the above two problems with delay (mn)^O(r) and (mn)^O(1), respectively, between consecutive outputs.

In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property 𝒫. Typically, the modification operation ⊠ may be vertex removal, edge removal, edge contraction, or edge addition and the question is, given a graph G and an integer k, whether it is possible to transform G to a graph in 𝒫 after applying k times the operation ⊠ on G. This problem has been extensively studied for particilar instantiations of ⊠ and 𝒫. In this paper we consider the general property 𝒫_ϕ of being planar and, moreover, being a model of some First-Order Logic sentence ϕ (an FOL-sentence). We call the corresponding meta-problem Graph ⊠-Modification to Planarity and ϕ and prove the following algorithmic meta-theorem: there exists a function f: ℕ² → ℕ such that, for every ⊠ and every FOL sentence ϕ, the Graph ⊠-Modification to Planarity and ϕ is solvable in f(k,|ϕ|)⋅n² time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman’s Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.

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.

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on 𝒪(k²) vertices.

We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. - For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known. - Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most m-k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most m-k arcs. We show that - Directed Multiplicative Spanner admits a polynomial kernel of size 𝒪(k⁴t⁵) and can be solved in randomized (4t)^k⋅ n^𝒪(1) time, - Directed Additive Spanner is W[1]-hard when parameterized by k even if t = 1 and the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is FPT when parameterized by t and k.

For a finite collection of graphs , the -TM-Deletion problem has as input an n-vertex graph G and an integer k and asks whether there exists a set S ⊆ V(G) with |S| ≤ k such that G\S does not contain any of the graphs in as a topological minor. We prove that for every such , -TM-Deletion is fixed parameter tractable on planar graphs. In particular, we provide an f(h, k) · n2 algorithm where h is an upper bound to the vertices of the graphs in .

Abstract For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. H-graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This class was introduced in the early 1990s by Bíró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Maximum Clique, Maximum Independent Set, or Minimum Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions. First we show that for every fixed H, the class of H-graphs is of logarithmically-bounded boolean-width (via mim-width). Pipelined with the plethora of known algorithms on graphs of bounded boolean-width, this describes a large class of problems solvable in polynomial time on H-graphs. We also observe that H-graphs are graphs with polynomially many minimal separators. Combined with the work of Fomin, Todinca and Villanger on algorithmic properties of such classes of graphs, this identify another wide class of problems solvable in polynomial time on H-graphs. The most fundamental optimization problems among the problems solvable in polynomial time on H-graphs are Maximum Clique, Maximum Independent Set, and Minimum Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Maximum Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, then Minimum Dominating Set is fixed-parameter tractable parameterized by the size of H. For Maximum Clique we show that it admits a polynomial kernel parameterized by H and the solution size.

Abstract A subgraph complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class $${\mathscr {G}}$$ G , is there a subgraph complement of G which is in $${\mathscr {G}}$$ G ? We show that this problem can be solved in polynomial time for various choices of the graphs class $${\mathscr {G}}$$ G , such as bipartite, d-degenerate, or cographs. We complement these results by proving that the problem is $${{\mathrm{NP}}}$$ NP -complete when $${\mathscr {G}}$$ G is the class of regular graphs.

Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the StrongF-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In StrongF-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence, the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to forbid that strong copy of F in G. We study StrongF-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when $$F =P_3$$ F=P3. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization $$k - \mu (G)$$ k-μ(G), where $$\mu (G)$$ μ(G) is the maximum matching size of G. We conclude with some results on the parameterization of StrongF-closure by the number of edges of G that are not selected as strong.

Low-rank binary matrix approximation is a generic problem where one seeks a good approximation of a binary matrix by another binary matrix with some specific properties. A good approximation means that the difference between the two matrices in some matrix norm is small. The properties of the approximation binary matrix could be: a small number of different columns, a small binary rank or a small Boolean rank. Unfortunately, most variants of these problems are NP-hard. Due to this, we initiate the systematic algorithmic study of low-rank binary matrix approximation from the perspective of parameterized complexity. We show in which cases and under what conditions the problem is fixed-parameter tractable, admits a polynomial kernel and can be solved in parameterized subexponential time.

Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. However, for many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded Π- Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties Π, even when parameterized by k + t . We show that the situation changes when we are looking for a connected induced subgraph satisfying Π. In particular, we show that the Connected Secluded Π -Subgraph problem is FPT when parameterized by just t for many important graph properties Π.

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 $(1+\varepsilon)d$.

We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constraints. The new constrained clustering problem generalizes a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are L ow GF(2)-R ank A pproximation , L ow B oolean -R ank A pproximation , and various versions of B inary C lustering . For example, for L ow GF(2)-R ank A pproximation problem, where for an m × n binary matrix A and integer r > 0, we seek for a binary matrix B of GF(2) rank at most r such that the ℓ 0 -norm of matrix A−B is minimum, our algorithm, for any ϵ > 0 in time f ( r ,ϵ)⋅ n ⋅ m , where f is some computable function, outputs a (1+ϵ)-approximate solution with probability at least (1−1\ e ). This is the first linear time approximation scheme for these problems. We also give (deterministic) PTASes for these problems running in time n f ( r )1\ϵ 2 log 1\ϵ , where f is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting on its own.

In k-Clustering we are given a multiset of n vectors X subset Z^d and a nonnegative number D, and we need to decide whether X can be partitioned into k clusters C_1, ..., C_k such that the cost sum_{i=1}^k min_{c_i in R^d} sum_{x in C_i} |x-c_i|_p^p <= D, where |*|_p is the Minkowski (L_p) norm of order p. For p=1, k-Clustering is the well-known k-Median. For p=2, the case of the Euclidean distance, k-Clustering is k-Means. We study k-Clustering from the perspective of parameterized complexity. The problem is known to be NP-hard for k=2 and it is also NP-hard for d=2. It is a long-standing open question, whether the problem is fixed-parameter tractable (FPT) for the combined parameter d+k. In this paper, we focus on the parameterization by D. We complement the known negative results by showing that for p=0 and p=infty, k-Clustering is W1-hard when parameterized by D. Interestingly, the complexity landscape of the problem appears to be more intricate than expected. We discover a tractability island of k-Clustering: for every p in (0,1], k-Clustering is solvable in time 2^O(D log D) (nd)^O(1).

Principal component analysis (PCA) is one of the most fundamental procedures in exploratory data analysis and is the basic step in applications ranging from quantitative finance and bioinformatics to image analysis and neuroscience. However, it is well-documented that the applicability of PCA in many real scenarios could be constrained by an "immune deficiency" to outliers such as corrupted observations. We consider the following algorithmic question about the PCA with outliers. For a set of $n$ points in $\mathbb{R}^{d}$, how to learn a subset of points, say 1% of the total number of points, such that the remaining part of the points is best fit into some unknown $r$-dimensional subspace? We provide a rigorous algorithmic analysis of the problem. We show that the problem is solvable in time $n^{O(d^2)}$. In particular, for constant dimension the problem is solvable in polynomial time. We complement the algorithmic result by the lower bound, showing that unless Exponential Time Hypothesis fails, in time $f(d)n^{o(d)}$, for any function $f$ of $d$, it is impossible not only to solve the problem exactly but even to approximate it within a constant factor.

We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In Clustering to Given Connectivities, we are given an n-vertex graph G, an integer k, and a sequence Lambda= of positive integers and we ask whether it is possible to remove at most k edges from G such that the resulting connected components are exactly t and their corresponding edge connectivities are lower-bounded by the numbers in Lambda. We prove that this problem, parameterized by k, is fixed parameter tractable, i.e., can be solved by an f(k)* n^{O(1)}-step algorithm, for some function f that depends only on the parameter k. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that we do not impose any restriction to the connectivity demands in Lambda.

A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems.

In the Edge Editing to Connected $f$-Degree Graph problem we are given a graph $G$, an integer $k$, and a function $f$ assigning integers to vertices of $G$. The task is to decide whether there is a connected graph $F$ on the same vertex set as $G$, such that for every vertex $v$, its degree in $F$ is $f(v)$, and the number of edges in $E(G)\triangle E(F)$, the symmetric difference of $E(G)$ and $E(F)$, is at most $k$. We show that Edge Editing to Connected $f$-Degree Graph is fixed-parameter tractable (FPT) by providing an algorithm solving the problem on an $n$-vertex graph in time $2^{\mathcal O(k)}n^{\mathcal O(1)}$. We complement this result by showing that the weighted version of the problem with costs $1$ and $0$ is W[1]-hard when parameterized by $k$ and the maximum value of $f$ even when the input graph is a tree. Our FPT algorithm is based on a nontrivial combination of color-coding and fast computations of representative families over the direct sum matroid of $\ell$-elongation of the co-graphic matroid associated with $G$ and a uniform matroid over the set of nonedges of $G$. We believe that this combination could be useful in designing parameterized algorithms for other edge editing and connectivity problems.

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid containing a set T of given terminal elements. For graphic matroids, this corresponds to the problem of finding a simple cycle passing through a set of given terminal edges in a graph. The algorithmic study of the problem on regular matroids, a superclass of graphic matroids, was initiated by Gavenčiak, Král’, and Oum [ICALP’12], who proved that the case of the problem with ∣T∣ = 2 is fixed-parameter tractable (FPT) when parameterized by the length of the circuit. We extend the result of Gavenčiak, Král’, and Oum by showing that for regular matroids • the M inimum S panning C ircuit problem, deciding whether there is a circuit with at most ℓ elements containing T , is FPT parameterized by k = ℓ − ∣T∣ • the S panning C ircuit problem, deciding whether there is a circuit containing ∣T∣, is FPT parameterized by ∣T∣. We note that extending our algorithmic findings to binary matroids, a superclass of regular matroids, is highly unlikely: M inimum S panning C ircuit parameterized by ℓ is W[1]-hard on binary matroids even when ∣T∣ = 1. We also show a limit to how far our results can be strengthened by considering a smaller parameter. More precisely, we prove that M inimum S panning C ircuit parameterized by ∣T∣ is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.

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 .

For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. These graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This notion was introduced in the early 1990s by Biro, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Clique, Independent Set, or Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions. First we show that for every fixed H, the class of H-graphs is of logarithmically-bounded boolean-width. We also prove that H-graphs are graphs with polynomially many minimal separators. Pipelined with the plethora of known algorithms on graphs of bounded boolean-width and graphs with polynomially many minimal separators, this describes a large class of optimization problems that are solvable in polynomial time on H-graphs. The most fundamental optimization problems among those solvable in polynomial time on H-graphs are Clique, Independent Set, and Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Independent Set and Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, Dominating Set is fixed-parameter tractable (FPT) parameterized by the size of H. Besides, we show that Clique admits a polynomial kernel parameterized by H and the solution size.

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.

A graph H is a square root of a graph G, or equivalently, G is the square of H, if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class $\mathcal {H}$ is called the $\mathcal {H}$ -Square Root problem. By showing boundedness of treewidth we prove that Square Root is polynomial-time solvable on some classes of graphs with small clique number and that $\mathcal {H}$ -Square Root is polynomial-time solvable when $\mathcal {H}$ is the class of cactuses.

Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in the design of parameterized algorithms on regular matroids. We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, in the Space Cover problem, we are given a matrix $M$ and a subset of its columns $T$; the task is to find a minimum set $F$ of columns of $M$ disjoint with $T$ such that the linear span of $F$ contains all vectors of $T$. For graphic matroids this problem is essentially Steiner Forest and for cographic matroids this is a generalization of Multiway Cut. Our main result is the algorithm with running time $2^{\mathcal{O}h(k)}\cdot ||M|| ^{\mathcal{O}h(1)}$ solving Space Cover in the case when $M$ is a totally unimodular matrix over rationals, where $k$ is the size of $F$. In other words, we show that on regular matroids the problem is fixed-parameter tractable parameterized by the rank of the covering subspace.

We initiate the algorithmic study of the following “structured augmentation” question: is it possible to increase the connectivity of a given graph $G$ by superposing it with another given graph $H$? More precisely, graph $F$ is the superposition of $G$ and $H$ with respect to injective mapping $\varphi\colon V(H)\rightarrow V(G)$ if every edge $uv$ of $F$ is either an edge of $G$ or $\varphi^{-1}(u)\varphi^{-1}(v)$ is an edge of $H.$ Thus $F$ contains both $G$ and $H$ as subgraphs, and the edge set of $F$ is the union of the edge sets of $G$ and $\varphi(H).$ We consider the following optimization problem. Given graphs $G,$ $H,$ and a weight function $\omega$ assigning nonnegative weights to pairs of vertices of $V(G),$ the task is to find $\varphi$ of minimum weight $\omega(\varphi)= \sum_{xy\in E(H)}\omega(\varphi(x)\varphi(y))$ such that the edge connectivity of the superposition $F$ of $G$ and $H$ with respect to $\varphi$ is higher than the edge connectivity of $G$. Our main result is the following “dichotomy” complexity classification. We say that a class of graphs $\mathcal{C}$ has bounded vertex-cover number if there is a constant $t$ depending on $\mathcal{C}$ only such that the vertex-cover number of every graph from $\mathcal{C}$ does not exceed $t.$ We show that for every class of graphs $\mathcal{C}$ with bounded vertex-cover number, the problems of superposing into a connected graph $F$ and to 2-edge connected graph $F$ are solvable in polynomial time when $H\in\mathcal{C}.$ On the other hand, for any hereditary class $\mathcal{C}$ with unbounded vertex-cover number, both problems are NP-hard when $H\in\mathcal{C}.$ For the unweighted variants of structured augmentation problems, i.e., the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems.

A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective $$H$$ -Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective $$H$$ -Colouring for every graph H on at most four vertices.

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid spanning a set t of given terminal elements. For graphic matroids this corresponds to the problem of finding a simple cycle passing through a set of given terminal edges in a graph. The algorithmic study of the problem on regular matroids, a superclass of graphic matroids, was initiated by Gavenčiak, Král', and Oum [ICALP'12], who proved that the case of the problem with |T| = 2 is fixed-parameter tractable (FPT) when parameterized by the length of the circuit. We extend the result of Gavenciak, Král', and Oum by showing that for regular matroids the Minimum Spanning Circuit problem, deciding whether there is a circuit with at most ℓ elements containing T, is FPT parameterized by k = ℓ — |T|; the Spanning Circuit problem, deciding whether there is a circuit containing T, is FPT parameterized by |T|. We note that extending our algorithmic findings to binary matroids, a superclass of regular matroids, is highly unlikely: Minimum Spanning Circuit parameterized by ℓ is W[1]- hard on binary matroids even when |T| = 1. We also show a limit to how far our results can be strengthened by considering a smaller parameter. More precisely, we prove that Minimum Spanning Circuit parameterized by |T| is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.

We consider the following graph modification problem. Let the input consist of a graph G = ( V , E ) , a weight function w : V ∪ E → N , a cost function c : V ∪ E → N 0 and a degree function δ : V → N 0 , together with three integers k v , k e and C . The question is whether we can delete a set of vertices of total weight at most k v and a set of edges of total weight at most k e so that the total cost of the deleted elements is at most C and every non-deleted vertex v has degree δ ( v ) in the resulting graph G ′ . We also consider the variant in which G ′ must be connected. Both problems are known to be NP -complete and W [ 1 ] -hard when parameterized by k v + k e . We prove that, when restricted to planar graphs, they stay NP -complete but have polynomial kernels when parameterized by k v + k e .

The notion of resolving sets in a graph was introduced by Slater [Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Util. Math., Winnipeg, 1975, pp. 549--559] and Harary and Melter [Ars Combin., 2 (1976), pp. 191--195] as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices $x$ and $y$ there is a vertex in the set which has distinct distances to $x$ and $y$. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud [Algorithmica, 2016, pp. 1--31] showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs, and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph.

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are at distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the “distance from triviality” framework. For an integer k , a planar + k v graph (or k -apex graph) is a graph that can be made planar by the removal of at most k vertices. We prove that a generalization of Square Root , in which some edges are prescribed to be either in or out of any solution, has a kernel of size O ( k ) for planar + k v graphs, when parameterized by k . Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

In the EDGE EDITING TO CONNECTED f-DEGREE GRAPH problem we are given a graph G, an integer k and a function f assigning integers to vertices of G. The task is to decide whether there is a connected graph F on the same vertex set as G, such that for every vertex v, its degree in F is f(v) and the number of edges inthe symmetric difference of E(G) and E(F), is at most k. We show that EDGE EDITING TO CONNECTED f-DEGREE GRAPH is fixed-parameter tractable (FPT) by providing an algorithm solving the problem on an n-vertex graph in time 2^{O(k)}n^{O(1)}. Our FPT algorithm is based on a non-trivial combination of color-coding and fast computations of representative families over direct sum matroid of l-elongation of co-graphic matroid associated with G and uniform matroid over the set of non-edges of G. We believe that this combination could be useful in designing parameterized algorithms for other edge editing problems.

The Secluded Path problem introduced by Chechik et al. in [ESA 2013] models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems. We start from an observation that being parameterized by the size of the exposure, the problem is fixed-parameter tractable (FPT). More precisely, we give an algorithm deciding if a graph G with a given cost function w:V(G)->N contains a secluded path of exposure at most k with the cost at most C in time O(3^{k/3}(n+m) log W), where W is the maximum value of w on an input graph G. Similarly, Secluded Steiner Tree is solvable in time O(2^{k}k^2 (n+m) log W). The main result of this paper is about "above guarantee" parameterizations for secluded problems. We show that Secluded Steiner Tree is FPT being parameterized by r+p, where p is the number of the terminals, l the size of an optimum Steiner tree, and r=k-l. We complement this result by showing that the problem is co-W[1]-hard when parameterized by r only. We also investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

The Plane Diameter Completion problem asks, given a plane graph G and a positive integer d, if it is a spanning subgraph of a plane graph H that has diameter at most d. We examine two variants of this problem where the input comes with another parameter k. In the first variant, called BPDC, k upper bounds the total number of edges to be added and in the second, called BFPDC, k upper bounds the number of additional edges per face. We prove that both problems are NP-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n^{3})+2^{2^{O((kd)^2\log d)}} * n steps.

The Hadwiger number of a graph $G$ is the largest integer $h$ such that $G$ has the complete graph $K_h$ as a minor. We show that the problem of determining the Hadwiger number of a graph is \sf NP-hard on co-bipartite graphs but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer $h$ such that $G$ has a minor with $h$ vertices and diameter at most $s$. We show that this problem can be solved in polynomial time on AT-free graphs when $s\geq 2$ but is \sf NP-hard on chordal graphs for every fixed $s\geq 2$.

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are said to be mutually induced if for any $1\leq i

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a nonnegative integer $k,$ decide whether the cyclability of $G$ is at least $k,$ is NP-hard. We study the parametrized complexity of this problem. We prove that this problem, parameterized by $k,$ is ${\sf co{-}W[1]}$-hard and that it does not admit a polynomial kernel on planar graphs, unless ${NP}\subseteq{\sf co}{-}{NP}/{poly}$. On the positive side, we give an FPT algorithm for planar graphs that runs in time $2^{2^{O(k^2\log k)}}\cdot n^2$. Our algorithm is based on a series of graph-theoretical results on cyclic linkages in planar graphs.

We study the paramereteized complexity of the following connectivity problem. For a vertex subset U of a graph G, trees T_1,...,T_s of G are completely independent spanning trees of U if each of them contains U, and for every two distinct vertices u,v in U, the paths from u to v in T_1,...,T_s are pairwise vertex disjoint except for end-vertices u and v. Then for a given s >= 2 and a parameter k, the task is to decide if a given n-vertex graph G contains a set U of size at least k such that there are s completely independent spanning trees of U. The problem is known to be NP-complete already for s=2. We prove the following results: (*) For s=2 the problem is solvable in time 2^{O(k)}*n^{O(1)}. (*) For s=2 the problem does not admit a polynomial kernel unless NP subseteq coNP/poly. (*) For arbitrary s, we show that the problem is solvable in time f(s,k)n^{O(1)} for some function f of s and k only.

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.

We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dominating Set problems on graphs of bounded clique-width. We show that on an $n$-vertex graph of clique-width $t$ both problems (1) cannot be solved in time $f(t)n^{o(t)}$ for any function $f$ of $t$ unless exponential time hypothesis fails, and (2) can be solved in time $n^{O(t)}$.

We study the parameterized complexity of the following Euler subgraph problems: (a) Large Euler Subgraph: For a given graph $G$ and integer parameter $k$, does $G$ contain an induced Eulerian subgraph with at least $k$ vertices? (b) Long Circuit: For a given graph $G$ and integer parameter $k$, does $G$ contain an Eulerian subgraph with at least $k$ edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. The complexity of the problem changes drastically on directed graphs, and we obtain the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed $k>3$ and is solvable in polynomial time for $k\leq 3$. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs.

The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. (ICALP '12) introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree k the equilibrium for this model is a maximal subgraph whose minimum degree is at least k. However the dropping out of individuals with degrees less than k might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. considered the Anchored k-Core problem: Given a graph G and integers b, k and p do there exist sets of vertices B, H such that B is a subset of H, size of B is at most b and size of H is at least p, and every vertex v which is in H but not in B has degree at least k in the induced subgraph G[H]. They showed that the problem is NP-hard for all k greater equal 2, and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the Anchored k-Core problem is W[1]-hard parameterized by p, even for k=3. This improves the result of Bhawalkar et al. (who show W[2]-hardness parameterized by b) as our parameter is always bigger since p is greater equal than b. Then we answer a question of Bhawalkar et al. by showing that the Anchored k-Core problem remains NP-hard on planar graphs for all k greater equal 3, even if the maximum degree of the graph is k+2. Finally we show that the problem is FPT on planar graphs parameterized by b for all k greater equal 7.

We consider the Directed Anchored k-Core problem, where the task is for a given directed graph G and integers b, k and p, to find an induced subgraph H with at least p vertices (the core) such that all but at most b vertices (the anchors) of H have in-degree at least k. For undirected graphs, this problem was introduced by Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012]. We undertake a systematic analysis of the computational complexity of Directed Anchored k-Core and show that: - The decision version of the problem is NP-complete for every k>=1 even if the input graph is restricted to be a planar directed acyclic graph of maximum degree at most k+2. - The problem is fixed parameter tractable (FPT) parameterized by the size of the core p for k=1, and W[1]-hard for k>=2. - When the maximum degree of the graph is at most Delta, the problem is FPT parameterized by p+Delta if k>=Delta/2.

Cai and Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs. For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about - a connected k-edge subgraph with all vertices of odd degrees, the problem known as k-Edge Connected Odd Subgraph; and - a connected k- vertex induced subgraph with all vertices of even degrees, the problem known as k-Vertex Eulerian Subgraph. We resolve both open problems and thus complete the characterization of even/odd subgraph problems from parameterized complexity perspective. We show that k-Edge Connected Odd Subgraph is FPT and that k-Vertex Eulerian Subgraph is W[1]-hard. Our FPT algorithm is based on a novel combinatorial result on the treewidth of minimal connected odd graphs with even amount of edges.

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k . We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d . In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k ≤3 the problem becomes polynomially time solvable.

Cops and robber is a classical pursuit-evasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem's complexity and combinatorial properties with constraining the following natural game parameters: fuel, the number of steps each cop can make; cost, the total sum of steps along edges all cops can make; and time, the number of rounds of the game.

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending the tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hypertree width of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph [Formula: see text] is constant factor sandwiched by the treewidth of its incidence graph when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above in which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some [Formula: see text]-minor-free graph class.

Many NP-hard problems can be solved efficiently when the input is restricted to graphs of bounded tree-width or clique-width. In particular, by the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the tree-width of the input graph. On the other hand if we restrict ourselves to graphs of clique-width at most t, then there are many natural problems for which the running time of the best known algorithms is of the form nf(t), where n is the input length and f is some function. It was an open question whether natural problems like Graph Coloring, Max-Cut, Edge Dominating Set, and Hamiltonian Path are fixed parameter tractable when parameterized by the clique-width of the input graph. As a first step toward obtaining lower bounds for clique-width parameterizations, in [SODA 2009], we showed that unless FPT≠W[1], there is no algorithm with run time O(g(t) · nc), for some function g and a constant c not depending on t, for Graph Coloring, Edge Dominating Set and Hamiltonian Path. But the lower bounds obtained in [SODA 2009] are weak when compared to the upper bounds on the time complexity of the known algorithms for these problems when parameterized by the clique-width. In this paper, we obtain the asymptotically tight bounds for Max-Cut and Edge Dominating Set by showing that both problems cannot be solved in time f(t)no(t), unless Exponential Time Hypothesis (ETH) collapses; and can be solved in time nO(t), where f is an arbitrary function of t, on input of size n and clique-width at most t. We obtain our lower bounds by giving non-trivial structure-preserving “linear FPT reductions”.

We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are $W[1]$-hard parameterized by clique-width. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, that is, solvable in time $g(k)\cdot n^{O(1)}$ on n-vertex graphs of clique-width k, where g is some function of k only. Our results imply that the running time $O(n^{f(k)})$ of many clique-width-based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely $FPT\neq W[1]$).

Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreover, for every fixed k ≥ 0, such problems can be solved in linear time on graphs of treewidth at most k. In particular, this implies that basic problems like Dominating Set, Graph Coloring, Clique, and Hamiltonian Cycle are solvable in linear time on graphs of bounded treewidth. A significant amount of research in graph algorithms has been devoted to extending this result to larger classes of graphs. It was shown that some of the algorithmic meta-theorems for treewidth can be carried over to graphs of bounded clique-width. Courcelle, Makowsky, and Rotics proved that the analogue of Courcelle's result holds for graphs of bounded clique-width when the logical formulas do not use edge set quantifications. Despite of its generality, this does not resolve the parameterized complexity of many basic problems concerning edge subsets (like Edge Dominating Set), vertex partitioning (like Graph Coloring), or global connectivity (like Hamiltonian Cycle). There are various algorithms solving some of these problems in polynomial time on graphs of clique-width at most k. However, these are not fixed parameter tractable algorithms and have typical running times O(nf(k)), where n is the input length and f is some function. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, i.e. solvable in time O(g(k)·nc), for some function g and a constant c not depending on k. In this paper we resolve this problem by showing that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W[1]-hard parameterized by clique-width. This shows that the running time O(nf(k)) of many clique-width based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely FPT ≠ W[1])—the price we pay for generality.

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello (PODS'99, PODS'01) in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx in SODA'06, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. Computing each of these width parameters is known to be an NP-hard problem. Moreover, the (generalized) hypertree width of an n-vertex hypergraph cannot be approximated within a logarithmic factor unless P=NP. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class. This extends the results of Feige, Hajiaghayi, and Lee from STOC'05 on approximating treewidth of H-minor-free graphs.

Inthe early studies on graph searching a graph was considered as a system of tunnels in which a fast and clever fugitive is hidden. The "classical" search problem is to find a search plan using the minimal number of searchers. In this paper, we consider a new criterion of optimization, namely, the search cost. First, we prove monotone properties of searching with the smallest cost. Then, making use of monotone properties, we prove that for any graph G the search cost of G is equal to the smallest number of edges of all interval supergraphs of G. Finally, we show how to compute the search cost of a cograph and the corresponding search strategy in linear time.