We study the problem of connectivity augmentation of a planar graph, while preserving planarity. This problem is motivated by many real-world settings such as road-networks, power-networks etc. In these settings, it is crucial to preserve the original planar embedding after augmentation. In 2009, Gutwenger and Mutzel gave a constructive algorithm showing that a connected planar graph with a fixed embedding (a plane graph) can be optimally augmented to a biconnected graph without crossings while preserving the embedding. We further this line of research, by giving an algorithm that computes a minimum set of edges that makes a connected plane graph 2-edge-connected in O(|V|(1+α(|V|))) time and linear space, where α is the inverse Ackermann function. We also study the 3-vertex-connectivity augmentation of biconnected outerplanar plane graphs. We present the first polynomial-time algorithm that augments such graphs to 3-connectivity with the minimum number of edges in O(|V|(1+α(|V|))) time and linear space while preserving the embedding, i.e. the augmented graph has a planar embedding that extends the given embedding.

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

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

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

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

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

In the Vertex Connectivity Survivable Network Design (VC-SNDP) problem, the input is a graph G and a function d: V(G) × V(G) → ℕ that encodes the vertex-connectivity demands between pairs of vertices. The objective is to find the smallest subgraph H of G that satisfies all these demands. It is a well-studied NP-complete problem that generalizes several network design problems. We consider the case of uniform demands, where for every vertex pair (u,v) the connectivity demand d(u,v) is a fixed integer κ. It is an important problem with wide applications. We study this problem in the realm of Parameterized Complexity. In this setting, in addition to G and d we are given an integer 𝓁 as the parameter and the objective is to determine if we can remove at least 𝓁 edges from G without violating any connectivity constraints. This was posed as an open problem by Bang-Jansen et.al. [SODA 2018], who studied the edge-connectivity variant of the problem under the same settings. Using a powerful classification result of Lokshtanov et al. [ICALP 2018], Gutin et al. [JCSS 2019] recently showed that this problem admits a (non-uniform) FPT algorithm where the running time was unspecified. Further they also gave an (uniform) FPT algorithm for the case of κ = 2. In this paper we present a (uniform) FPT algorithm any κ that runs in time 2^{O(κ² 𝓁⁴ log 𝓁)}⋅ |V(G)|^O(1). Our algorithm is built upon new insights on vertex connectivity in graphs. Our main conceptual contribution is a novel graph decomposition called the Wheel decomposition. Informally, it is a partition of the edge set of a graph G, E(G) = X₁ ∪ X₂ … ∪ X_r, with the parts arranged in a cyclic order, such that each vertex v ∈ V(G) either has edges in at most two consecutive parts, or has edges in every part of this partition. The first kind of vertices can be thought of as the rim of the wheel, while the second kind form the hub. Additionally, the vertex cuts induced by these edge-sets in G have highly symmetric properties. Our main technical result, informally speaking, establishes that "nearly edge-minimal’’ κ-vertex connected graphs admit a wheel decomposition - a fact that can be exploited for designing algorithms. We believe that this decomposition is of independent interest and it could be a useful tool in resolving other open problems.

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

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

Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining time algorithms for a family of graph modification problems that includes problems that can be seen as generalized cycle hitting problems. Our starting point is the Node Unique Label Cover problem (that is, given a CSP instance where each constraint is a permutation of values on two variables, the task is to delete k variables to make the instance satisfiable). We introduce a variant of the problem where k vertices have to be deleted such that every 2-connected component of the remaining instance is satisfiable. Then we extend the problem with cardinality constraints that restrict the number of times a certain value can be used (globally or within a 2-connected component of the solution). We show that there is an time algorithm on planar graphs for any problem that can be formulated this way, which includes a large number of well-studied problems, for example, Odd Cycle Transversal, Subset Feedback Vertex Set, Group Feedback Vertex Set, Subset Group Feedback Vertex Set, Vertex Multiway Cut, and Component Order Connectivity. For those problems that admit appropriate (quasi)polynomial kernels (that increase the parameter only linearly and preserve planarity), our results immediately imply time parameterized algorithms on planar graphs. In particular, we use or adapt known kernelization results to obtain time (randomized) algorithms for Vertex Multiway Cut, Group Feedback Vertex Set, and Subset Feedback Vertex Set. Our algorithms are designed with possible generalization to H-minor free graphs in mind. To obtain the same time algorithms on H-minor free graphs, the only missing piece is the vertex version of a contraction decomposition theorem that we currently have only for planar graphs.

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.

We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envy-freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of EFX allocations has not been settled and is one of the most important problems in fair division [5]. Towards resolving this problem, many impressive results show the existence of its relaxations. In particular, [1] shows the existence of 0.618-EFX allocations, and [4] shows that EFX allocation exists if we do not allocate at most n - 1 goods. The latter result was recently improved for three agents in [2], in which the two unallocated goods are allocated through an involved procedure. Reducing the number of unallocated goods for an arbitrary number of agents is a systematic way to settle the big question.

Augmenting an undirected or a directed graph (digraph) by adding new edges or arcs, to increase its connectivity to a target value, is a fundamental problem in combinatorial optimization and graph theory. In this paper we study the basic problem of augmenting an input digraph to make it strongly connected, which is known as the Strong Connectivity Augmentation problem. Here, the input is a digraph D = (V, A), a set of links L ⊆ V × V, and a positive integer k. The objective is to decide if there exists a subset F ⊆ L, of size at most k, such that D′ = (V, A ∪ F) is strongly connected. We consider the general version of this problem where, additionally, there is a weight function w : L → ℝ+ on the links, and the goal is to find a minimum weight subset F ⊆ L of cardinality at most k, such that D′ = (V, A ∪ F) is strongly connected. We design an algorithm for this problem that runs in time 2(k log k) n(1), thereby showing that it is fixed parameter tractable (FPT). Here, n = |V|. This also resolves an open problem stated by Guo and Uhlmann more than a decade ago [Networks 56(2): 131–142 (2010)].

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

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

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 the Split Vertex Deletion (SVD) problem, the input is an n-vertex undirected graph G and a weight function w: V(G) → ℕ, and the objective is to find a minimum weight subset S of vertices such that G-S is a split graph (i.e., there is bipartition of V(G-S) = C ⊎ I such that C is a clique and I is an independent set in G-S). This problem is a special case of 5-Hitting Set and consequently, there is a simple factor 5-approximation algorithm for this. On the negative side, it is easy to show that the problem does not admit a polynomial time (2-δ)-approximation algorithm, for any fixed δ > 0, unless the Unique Games Conjecture fails. We start by giving a simple quasipolynomial time (n^O(log n)) factor 2-approximation algorithm for SVD using the notion of clique-independent set separating collection. Thus, on the one hand SVD admits a factor 2-approximation in quasipolynomial time, and on the other hand this approximation factor cannot be improved assuming UGC. It naturally leads to the following question: Can SVD be 2-approximated in polynomial time? In this work we almost close this gap and prove that for any ε > 0, there is a n^O(log 1/(ε))-time 2(1+ε)-approximation algorithm.

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

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.

In this paper we study two classical cut problems, namely Multicut and Multiway Cut on chordal graphs and split graphs. In the Multicut problem, the input is a graph G, a collection of 𝓁 vertex pairs (s_i, t_i), i ∈ [𝓁], and a positive integer k and the goal is to decide if there exists a vertex subset S ⊆ V(G)⧵ {s_i,t_i : i ∈ [𝓁]} of size at most k such that for every vertex pair (s_i,t_i), s_i and t_i are in two different connected components of G-S. In Unrestricted Multicut, the solution S can possibly pick the vertices in the vertex pairs {(s_i,t_i): i ∈ [𝓁]}. An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct vertices in T× T. The fixed parameter tractability (FPT) of these problems was a long-standing open problem and has been resolved fairly recently. Multicut and Multiway Cut now admit algorithms with running times 2^{{𝒪}(k³)}n^{{𝒪}(1)} and 2^k n^{{𝒪}(1)}, respectively. However, the kernelization complexity of both these problems is not fully resolved: while Multicut cannot admit a polynomial kernel under reasonable complexity assumptions, it is a well known open problem to construct a polynomial kernel for Multiway Cut. Towards designing faster FPT algorithms and polynomial kernels for the above mentioned problems, we study them on chordal and split graphs. In particular we obtain the following results. 1) Multicut on chordal graphs admits a polynomial kernel with {𝒪}(k³ 𝓁⁷) vertices. Multiway Cut on chordal graphs admits a polynomial kernel with {𝒪}(k^{13}) vertices. 2) Multicut on chordal graphs can be solved in time min {𝒪(2^{k} ⋅ (k³+𝓁) ⋅ (n+m)), 2^{𝒪(𝓁 log k)} ⋅ (n+m) + 𝓁 (n+m)}. Hence Multicut on chordal graphs parameterized by the number of terminals is in XP. 3) Multicut on split graphs can be solved in time min {𝒪(1.2738^k + kn+𝓁(n+m), 𝒪(2^{𝓁} ⋅ 𝓁 ⋅ (n+m))}. Unrestricted Multicut on split graphs can be solved in time 𝒪(4^{𝓁}⋅ 𝓁 ⋅ (n+m)).

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T – S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here the input is a tournament T and a weight function w: V(T) → ℕ and the task is to find a feedback vertex set S in T minimizing w(S) = ΣvϵSw(v). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this paper we give the first polynomial time factor 2 approximation algorithm for this problem. Assuming the Unique Games conjecture, this is the best possible approximation ratio achievable in polynomial time.

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

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

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

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

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

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

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

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

A generalization of classical cycle hitting problems, called conflict version of the problem, is defined as follows. An input is undirected graphs G and H on the same vertex set, and a positive integer k, and the objective is to decide whether there exists a vertex subset X subseteq V(G) such that it intersects all desired "cycles" (all cycles or all odd cycles or all even cycles) and X is an independent set in H. In this paper we study the conflict version of classical Feedback Vertex Set, and Odd Cycle Transversal problems, from the view point of kernelization complexity. In particular, we obtain the following results, when the conflict graph H belongs to the family of d-degenerate graphs. 1) CF-FVS admits a O(k^{O(d)}) kernel. 2) CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP subseteq coNP/poly. For our kernelization algorithm we exploit ideas developed for designing polynomial kernels for the classical Feedback Vertex Set problem, as well as, devise new reduction rules that exploit degeneracy crucially. Our main conceptual contribution here is the notion of "k-independence preserver". Informally, it is a set of "important" vertices for a given subset X subseteq V(H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size k^{O(d)} exists, and can be used in designing polynomial kernel.

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

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

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

In the Subset Odd Cycle Transversal (Subset OCT) problem, the input is a graph $G$, a subset of vertices $T$ and a positive integer $k$ and the objective is to determine whether there exists a $k$-sized vertex subset that intersects every odd cycle containing a vertex from $T$. Clearly, Subset OCT is a generalization of the classic Odd Cycle Transversal problem where the objective is to determine whether there exists a $k$-sized vertex subset that intersects every odd cycle in the given graph. We remark that Subset OCT also generalizes the well known Multiway Cut problem, as well as a parity constrained variant, the Odd Multiway Cut problem. Recently, Kakimura, Kawarabayashi, and Kobayashi [ Proceedings of SODA, 2012, pp. 1726--1736] proposed a fixed parameter tractable (FPT) algorithm for this problem that runs in time $f(k)mn^3$ using the theory of graph minors, where $f$ is some function, and $n$ and $m$ denote the number of vertices and edges in the graph. However, the dependence of this function on $k$ is at least triple exponential. In this paper, we give the first FPT algorithm for this problem where the exponential dependence of the running time of the algorithm on $k$ is polynomial. Our algorithm avoids the use of the theory of graph minors, is self contained, and runs in time $2^{\mathcal{O}(k^3\log k)}mn^2\log^2 n$, thus improving upon the algorithm of Kakimura and co-authors with respect to both the parameter as well as the input size. Our algorithm utilizes a recursive application of “generalized” important separators to reduce the subset version of this problem to the standard version of the problem.

We study some well-known graph contraction problems in the recently introduced framework of lossy kernelization. In classical kernelization, given an instance (I,k) of a parameterized problem, we are interested in obtaining (in polynomial time) an equivalent instance (I',k') of the same problem whose size is bounded by a function in k. This notion however has a major limitation. Given an approximate solution to the instance (I',k'), we can say nothing about the original instance (I,k). To handle this issue, among others, the framework of lossy kernelization was introduced. In this framework, for a constant alpha, given an instance (I,k) we obtain an instance (I',k') of the same problem such that, for every c>1, any c-approximate solution to (I',k') can be turned into a (c*alpha)-approximate solution to the original instance (I, k) in polynomial time. Naturally, we are interested in a polynomial time algorithm for this task, and further require that |I'| + k' = k^{O(1)}. Akin to the notion of polynomial time approximation schemes in approximation algorithms, a parameterized problem is said to admit a polynomial size approximate kernelization scheme (PSAKS) if it admits a polynomial size alpha-approximate kernel for every approximation parameter alpha > 1. In this work, we design PSAKSs for Tree Contraction, Star Contraction, Out-Tree Contraction and Cactus Contraction problems. These problems do not admit polynomial kernels, and we show that each of them admit a PSAKS with running time k^{f(alpha)}|I|^{O(1)} that returns an instance of size k^{g(alpha)} where f(alpha) and g(alpha) are constants depending on alpha.

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on $n$ vertices and a positive integer parameter $k$, find if there exist $k$ edges (arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the vertices have even degrees---where the resulting graph is \emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica, 2014}] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time $2^{O(k \log k)}n^{O(1)}$. They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. In this paper we answer their question in the affirmative: using the technique of computing \emph{representative families of co-graphic matroids} we design algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity.

Let $$M=(E,\mathcal{I})$$ be a matroid. A $$k$$ -truncation of $$M$$ is a matroid $$M'=(E,\mathcal{I}')$$ such that for any $$A\subseteq E$$ , $$A\in \mathcal{I}'$$ if and only if $$|A|\le k$$ and $$A\in \mathcal {I}$$ . Given a linear representation of $$M$$ we consider the problem of finding a linear representation of the $$k$$ -truncation of this matroid. This problem can be expressed as the following problem on matrices. Let $$M$$ be a $$n\times m$$ matrix over a field $$\mathbb {F}$$ . A rank $$k$$ -truncation of the matrix $$M$$ is a $$k\times m$$ matrix $$M_k$$ (over $${\mathbb F}$$ or a related field) such that for every subset $$I\subseteq \{1,\ldots ,m\}$$ of size at most $$k$$ , the set of columns corresponding to $$I$$ in $$M$$ has rank $$|I|$$ if and only if the corresponding set of columns in $$M_k$$ has rank $$|I|$$ . A common way to compute a rank $$k$$ -truncation of a $$n \times m$$ matrix is to multiply the matrix with a random $$k\times n$$ matrix (with the entries from a field of an exponential size), yielding a simple randomized algorithm. So a natural question is whether it possible to obtain a rank $$k$$ -truncation of a matrix, deterministically. In this paper we settle this question for matrices over any field in which the field operations can be done efficiently. This includes any finite field and the field of rationals ( $$\mathbb Q$$ ). Our algorithms are based on the properties of the classical Wronskian determinant, and the folded Wronskian determinant, which was recently introduced by Guruswami and Kopparty [ FOCS, 2013 ], and was implicitly present in the work of Forbes and Shpilka [ STOC, 2012 ]. These were used in the context of subspace designs, and reducing randomness for polynomial identity testing and other related problems. Our main conceptual contribution in this paper is to show that the Wronskian determinant can also be used to obtain a representation of the truncation of a linear matroid in deterministic polynomial time. Finally, we use our results to derandomize several parameterized algorithms, including an algorithm for computing $$\ell $$ -Matroid Parity, to which several problems like $$\ell $$ -Matroid Intersection can be reduced.