Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

Parameterized Approximation Schemes using Graph Widths

Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, evading both polynomial-time inapproximability and parameterized intractability bounds

More applications of the d-neighbor equivalence: acyclicity and connectivity constraints

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that solves all these problems and whose running time is upper bounded by $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ and $n^{O(k)}$ where $k$ is respectively the clique-width, $\mathbb{Q}$-rank-width, rank-width and maximum induced matching width of a given decomposition. Our meta-algorithm simplifies and unifies the known algorithms for each of the parameters and its running time matches asymptotically also the running times of the best known algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the $d$-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance of this equivalence relation on the algorithmic applications of width measures. We also prove that our framework could be useful for $W[1]$-hard problems parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For these latter problems, we obtain $n^{O(k)}$, $n^{O(k)}$ and $n^{2^{O(k)}}$ time algorithms where $k$ is respectively the clique-width, the $\mathbb{Q}$-rank-width and the rank-width of the input graph

On the Equivalence among Problems of Bounded Width

In this paper, we introduce a methodology, called decomposition-based reductions, for showing the equivalence among various problems of bounded-width. First, we show that the following are equivalent for any $\alpha > 0$: * SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * 3-SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * Max 2-SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * Independent Set can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, and * Independent Set can be solved in $O^*(2^{\alpha \mathrm{cw}})$ time, where tw and cw are the tree-width and clique-width of the instance, respectively. Then, we introduce a new parameterized complexity class EPNL, which includes Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and Independent Set parameterized by path-width are EPNL-complete. This implies that if one of these EPNL-complete problems can be solved in $O^*(c^k)$ time, then any problem in EPNL can be solved in $O^*(c^k)$ time.Comment: accepted to ESA 201