88 research outputs found

    Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?

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    We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Transversal, Chordal Deletion, Eta Transversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like F-Minor-Free Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F. While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing, which we find surprising. While it was known that testing for an induced complete subgraph has no polynomial kernel unless NP is in coNP/poly, we show that the problem of testing if a graph contains a complete graph on t vertices as a minor admits a polynomial kernel. On the other hand, it was known that testing for a path on t vertices as a minor admits a polynomial kernel, but we show that testing for containment of an induced path on t vertices is unlikely to admit a polynomial kernel.Comment: To appear in the Journal of Computer and System Science

    Kernelization Lower Bounds By Cross-Composition

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    We introduce the cross-composition framework for proving kernelization lower bounds. A classical problem L AND/OR-cross-composes into a parameterized problem Q if it is possible to efficiently construct an instance of Q with polynomially bounded parameter value that expresses the logical AND or OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem OR-cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy collapses. Similarly, an AND-cross-composition for Q rules out polynomial kernels for Q under Bodlaender et al.'s AND-distillation conjecture. Our technique generalizes and strengthens the recent techniques of using composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. After learning of our results, several teams of authors have successfully applied the cross-composition framework to different parameterized problems. For completeness, our presentation of the framework includes several extensions based on this follow-up work. For example, we show how a relaxed version of OR-cross-compositions may be used to give lower bounds on the degree of the polynomial in the kernel size.Comment: A preliminary version appeared in the proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) under the title "Cross-Composition: A New Technique for Kernelization Lower Bounds". Several results have been strengthened compared to the preliminary version (http://arxiv.org/abs/1011.4224). 29 pages, 2 figure

    Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations

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    In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by mm disjoint sets V1,V2,,VmV^1, V^2, \dots, V^m, each composed of nn binary vectors of size pp. An output is a set of nn disjoint mm-tuples of vectors, where each mm-tuple is obtained by picking one vector from each set ViV^i. To each mm-tuple we associate a pp dimensional vector by applying the bit-wise AND operation on the mm vectors of the tuple. The objective is to minimize the total number of zeros in these nn vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results (ETHETH-based results, WW[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter kk i.e. the total number of zeros), as well as two parameters above some guaranteed values.Comment: 16 pages, 6 figure

    Structural Parameterizations with Modulator Oblivion

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    It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph, when parameterized by k. While this investigation fits naturally into the recent trend of what are called "structural parameterizations", here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least k^O(k)n^O(1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2^O(k)n^O(1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in time 2^O(k)n^O(1) where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the Strong Exponential Time Hypothesis (SETH)

    Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

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    We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G,k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class ?, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes ? for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to ?, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families ? for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if ? has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number

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

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    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\u27,S\u27,l\u27) such that |V(G\u27)|<= O(k^5), |E(G\u27)|<= O(k^6) and G has a vertex cover of size at most l if and only if G\u27 has a vertex cover of size at most l\u27. 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

    Smaller Parameters for Vertex Cover Kernelization

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    We revisit the topic of polynomial kernels for Vertex Cover relative to structural parameters. Our starting point is a recent paper due to Fomin and Str{\o}mme [WG 2016] who gave a kernel with O(X12)\mathcal{O}(|X|^{12}) vertices when XX is a vertex set such that each connected component of GXG-X contains at most one cycle, i.e., XX is a modulator to a pseudoforest. We strongly generalize this result by using modulators to dd-quasi-forests, i.e., graphs where each connected component has a feedback vertex set of size at most dd, and obtain kernels with O(X3d+9)\mathcal{O}(|X|^{3d+9}) vertices. Our result relies on proving that minimal blocking sets in a dd-quasi-forest have size at most d+2d+2. This bound is tight and there is a related lower bound of O(Xd+2ϵ)\mathcal{O}(|X|^{d+2-\epsilon}) on the bit size of kernels. In fact, we also get bounds for minimal blocking sets of more general graph classes: For dd-quasi-bipartite graphs, where each connected component can be made bipartite by deleting at most dd vertices, we get the same tight bound of d+2d+2 vertices. For graphs whose connected components each have a vertex cover of cost at most dd more than the best fractional vertex cover, which we call dd-quasi-integral, we show that minimal blocking sets have size at most 2d+22d+2, which is also tight. Combined with existing randomized polynomial kernelizations this leads to randomized polynomial kernelizations for modulators to dd-quasi-bipartite and dd-quasi-integral graphs. There are lower bounds of O(Xd+2ϵ)\mathcal{O}(|X|^{d+2-\epsilon}) and O(X2d+2ϵ)\mathcal{O}(|X|^{2d+2-\epsilon}) for the bit size of such kernels
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