Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial O(n k ) time brute force algorithm for finding a k-element solution, we try to give algorithms with uniformly polynomial (i.e., f(k) n 0(1) ) running time. The main result is that if the ground set of a represented matroid is partitioned into blocks of size l, then we can determine in f(k, (l). n 0(1 randomized time whether there is an independent set that is the union of k blocks. As consequence, algorithms with similar running time are obtained for other problems such as finding a κ-set in the intersection of l matroids, or finding κ terminals in a network such that each of them can be connected simultaneously to the source by l disjoint paths.

A Parameterized View on Matroid Optimization Problems

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems More recently, the two have also been combined to derive many interesting results We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · nO(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · nO(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. The first class of parameterizations is based on the notion of elimination distance of the input graph to the target graph class , which intuitively measures the number of rounds needed to obtain a graph in by removing one vertex from each connected component in each round. The second class of parameterizations consists of a relaxation of the notion of treewidth, allowing arbitrarily large bags that induce subgraphs belonging to the target class of the deletion problem as long as these subgraphs have small neighborhoods. Both kinds of parameterizations have been introduced recently and have already spawned several independent results. Our contribution is twofold. First, we present a framework for computing approximately optimal decompositions related to these graph measures. Namely, if the cost of an optimal decomposition is k, we show how to find a decomposition of cost kO(1) in time f(k) · nO(1). This is applicable to any class for which we can solve the so-called separation problem. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations. For example, we present an algorithm running in time nO(1) + 2kO(1)·(n+m) and polynomial space for Odd cycle transversal parameterized by the elimination distance k to the class of bipartite graphs.

/pdf/vertex-deletion-parameterized-by-elimination-distance-and-14mtrrl7as.pdf

Vertex deletion parameterized by elimination distance and even less

The organizers of the 4th Parameterized Algorithms and Computational Experiments challenge (PACE 2019) report on the 4th iteration of the PACE challenge. This year, the first track featured the MinVertexCover problem, which asks given an undirected graph G=(V,E) to output a set S subseteq V of vertices such that for every edge vw in E at least one endpoint belongs to S. The exact decision version of this problem is one of the most discussed problem if not even the prototypical problem in parameterized complexity theory. Another two tracks were dedicated to computing the hypertree width of a given hypergraph, which is a certain generalization of tree decompositions to hypergraphs that has widely been applied to problems in databases, constraint programming, and artificial intelligence. On one track we asked for submissions that compute hypertree decompositions of minimum width (MinHypertreeWidth) and on the other track we asked to heuristically compute hypertree decompositions of small width quickly (HeurHypertreeWidth). We received 28 implementations from 26 teams. This year we asked participants to submit solver descriptions in order to count as a submission for the challenge. We received those from 16 teams with overall 33 participants from 10 countries. One team submitted successful solutions to all three tracks.

https://drops.dagstuhl.de/opus/volltexte/2019/11486/pdf/LIPIcs-IPEC-2019-25.pdf/

The PACE 2019 Parameterized Algorithms and Computational Experiments Challenge: The Fourth Iteration (Invited Paper)

Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms. It admits a kernel with $\phantom {\dot {i}\!}\mathcal {O}(\ell ^{2})$
 edges and $\phantom {\dot {i}\!}2\ell $
 vertices where $\phantom {\dot {i}\!}\ell $
 denotes the size of the vertex cover we are seeking for. A natural question is whether Vertex Cover is fixed-parameter tractable or admits a polynomial kernel with respect to a parameter k, that is, provably smaller than the size of the vertex cover. Jansen and Bodlaender [STACS 2011, TOCS 2013] raised this question and gave a kernel for Vertex Cover of size $\phantom {\dot {i}\!}\mathcal {O}(f^{3})$
 , where f is the size of a feedback vertex set of the input graph. We continue this line of work and study Vertex Cover with respect to a parameter that is always smaller than the solution size and incomparable to the size of the feedback vertex set of the input graph. Our parameter is the number of vertices whose removal results in a graph of maximum degree two. While vertex cover with this parameterization can easily be shown to be fixed-parameter tractable (FPT), we show that it has a polynomial kernel. The input to our problem consists of an undirected graph G, $\phantom {\dot {i}\!}S \subseteq V(G)$
 such that $\phantom {\dot {i}\!}|S| = k$
 and $\phantom {\dot {i}\!}G[V(G)\setminus S]$
 has maximum degree at most two and a positive integer $\phantom {\dot {i}\!}\ell $
 . Given $\phantom {\dot {i}\!}(G,S,\ell )$
 , in polynomial time we output an instance $\phantom {\dot {i}\!}(G^{\prime },S^{\prime },\ell ^{\prime })$
 such that $\phantom {\dot {i}\!}|V(G^{\prime })|$
 is $\phantom {\dot {i}\!}\mathcal {O}(k^{5})$
 , $\phantom {\dot {i}\!}|E(G^{\prime })|$
 is $\phantom {\dot {i}\!}\mathcal {O}(k^{6})$
 and G has a vertex cover of size at most $\phantom {\dot {i}\!}\ell $
 if and only if $\phantom {\dot {i}\!}G^{\prime }$
 has a vertex cover of size at most $\phantom {\dot {i}\!}\ell ^{\prime }$
 . When $\phantom {\dot {i}\!}G[V(G)\setminus S]$
 has maximum degree at most one, we improve the known kernel bound from $\phantom {\dot {i}\!}\mathcal {O}(k^{3})$
 vertices to $\phantom {\dot {i}\!}\mathcal {O}(k^{2})$
 vertices (and $\phantom {\dot {i}\!}\mathcal {O}(k^{3})$
 edges). More generally, given $\phantom {\dot {i}\!}(G, S, \ell )$
 where every connected component of $\phantom {\dot {i}\!}G \setminus S$
 is a clique of at most d vertices (for constant d), in polynomial time, we output an equivalent instance $\phantom {\dot {i}\!}(G^{\prime }, S^{\prime }, \ell ^{\prime })$
 for the same problem where $\phantom {\dot {i}\!}|V(G^{\prime })|$
 is $\phantom {\dot {i}\!}\mathcal {O}(k^{d})$
 . We also show that for this problem, when $\phantom {\dot {i}\!}d \geq 3$
 , a kernel with $\phantom {\dot {i}\!}\mathcal {O}(k^{d-\varepsilon })$
 bits cannot exist for any $\phantom {\dot {i}\!}\varepsilon >0$
 unless $\phantom {\dot {i}\!}\textsf {NP} \subseteq \textsf {coNP}/\textsf {poly}$
 .

Polynomial Kernels for Vertex Cover Parameterized by Small Degree Modulators

The Connected Vertex Cover problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al. [STOC 2017], have shown an α-approximate kernel for the problem for every α > 1, in the framework of approximate or lossy kernelization. In this work, we exhibit lossy kernels and FPT algorithms for Connected Vertex Cover for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. Our first result is a lossy kernel for Connected Vertex Cover parameterized by the size of a split deletion set. Let n denote the number of vertices in the input graph. We show that To the best of our knowledge, this (approximate) kernel is one of the few lossy kernels for problems parameterized by a structural parameter (that is not solution size). We extend this lossy kernelization to Connected Vertex Cover parameterized by an incomparable parameter, and that is the size of a clique cover. We show that Finally, we consider the size of a cluster deletion set as parameter. We show that

pdf/revisiting-connected-vertex-cover-fpt-algorithms-and-lossy-u4li2jvw1l.pdf

Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms and admits a kernel with O(l^2) edges and 2*l vertices. Here, l denotes the size of a vertex cover we are seeking for. A natural question is whether Vertex Cover admits a polynomial kernel (or a parameterized algorithm) with respect to a parameter k, that is, provably smaller than the size of the vertex cover. Jansen and Bodlaender [STACS 2011, TOCS 2013] raised this question and gave a kernel for Vertex Cover of size O(f^3), where f is the size of a feedback vertex set of the input graph. We continue this line of work and study Vertex Cover with respect to a parameter that is always smaller than the solution size and incomparable to the size of the feedback vertex set of the input graph. Our parameter is the number of vertices whose removal results in a graph of maximum degree two. While vertex cover with this parameterization can easily be shown to be fixed-parameter tractable (FPT), we show that it has a polynomial sized kernel. 

The input to our problem consists of an undirected graph G, S \subseteq V(G) such that |S| = k and G[V(G)\S] has maximum degree at most 2 and a positive integer l. Given (G,S,l), in polynomial time we output an instance (G',S',l') such that |V(G')|<= O(k^5), |E(G')|<= O(k^6) and G has a vertex cover of size at most l if and only if G' has a vertex cover of size at most l'. 
When G[V(G)\S] has maximum degree at most 1, we improve the known kernel bound from O(k^3) vertices to O(k^2) vertices (and O(k^3) edges). In general, if G[V(G)\S] is simply a collection of cliques of size at most d, then we transform the graph in polynomial time to an equivalent hypergraph with O(k^d) vertices and show that, for d >= 3, a kernel with O(k^{d-epsilon}) vertices is unlikely to exist for any epsilon >0 unless NP is a subset of coNO/poly.

Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators

A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure in the input. In particular, we consider parameterizations where the parameter is (instead of the solution size), the distance to a class of graphs where the problem is polynomial time solvable, and sometimes smaller than the solution size. Here, by distance to a class of graphs, we mean the minimum number of vertices whose removal results in a graph in the class. Such a set of vertices is also called the ‘deletion set’. In this paper, we show that
 Regarding kernelization results, we show that

Structural Parameterizations of Undirected Feedback Vertex Set: FPT Algorithms and Kernelization

Set Cover is one of the well-known classical NP-hard problems. Following some recent trends, we study the conflict-free version of the Set Cover problem. Here we have a universe ${\mathcal U}$, a family ${\mathcal F}$ of subsets of ${\mathcal U}$ and a graph $G_{{\mathcal F}}$ on the vertex set ${\mathcal F}$ and we look for a subfamily ${\mathcal F}' \subseteq {\mathcal F}$ of minimum size that covers ${\mathcal U}$ and also forms an independent set in $G_{{\mathcal F}}$. Here we initiate a systematic study of the problem in parameterized complexity by restricting the focus to the variants where Set Cover is fixed-parameter tractable (FPT). We give upper bounds and lower bounds for conflict-free version of the Set Cover with and without duplicate sets along with restrictions to the graph classes of $G_{{\mathcal F}}$.

Parameterized Complexity of Conflict-Free Set Cover

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