16,335 research outputs found

    The node-deletion problem for hereditary properties is NP-complete

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    AbstractWe consider the family of graph problems called node-deletion problems, defined as follows; For a fixed graph property Π, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies Π? We show that if Π is nontrivial and hereditary on induced subgraphs, then the node-deletion problem for Π is NP-complete for both undirected and directed graphs

    On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion

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    In this paper, we investigate the approximability of two node deletion problems. Given a vertex weighted graph G=(V,E)G=(V,E) and a specified, or "distinguished" vertex pVp \in V, MDD(min) is the problem of finding a minimum weight vertex set SV{p}S \subseteq V\setminus \{p\} such that pp becomes the minimum degree vertex in G[VS]G[V \setminus S]; and MDD(max) is the problem of finding a minimum weight vertex set SV{p}S \subseteq V\setminus \{p\} such that pp becomes the maximum degree vertex in G[VS]G[V \setminus S]. These are known NPNP-complete problems and have been studied from the parameterized complexity point of view in previous work. Here, we prove that for any ϵ>0\epsilon > 0, both the problems cannot be approximated within a factor (1ϵ)logn(1 - \epsilon)\log n, unless NPDTIME(nloglogn)NP \subseteq DTIME(n^{\log\log n}). We also show that for any ϵ>0\epsilon > 0, MDD(min) cannot be approximated within a factor (1ϵ)logn(1 -\epsilon)\log n on bipartite graphs, unless NPDTIME(nloglogn)NP \subseteq DTIME(n^{\log\log n}), and that for any ϵ>0\epsilon > 0, MDD(max) cannot be approximated within a factor (1/2ϵ)logn(1/2 - \epsilon)\log n on bipartite graphs, unless NPDTIME(nloglogn)NP \subseteq DTIME(n^{\log\log n}). We give an O(logn)O(\log n) factor approximation algorithm for MDD(max) on general graphs, provided the degree of pp is O(logn)O(\log n). We then show that if the degree of pp is nO(logn)n-O(\log n), a similar result holds for MDD(min). We prove that MDD(max) is APXAPX-complete on 3-regular unweighted graphs and provide an approximation algorithm with ratio 1.5831.583 when GG is a 3-regular unweighted graph. In addition, we show that MDD(min) can be solved in polynomial time when GG is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete Algorithm

    Parameterized Complexity Dichotomy for Steiner Multicut

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    The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt \subseteq V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). We present two proofs: one using the randomized contractions technique of Chitnis et al, and one relying on new structural lemmas that decompose the Steiner cut into important separators and minimal s-t cuts. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).Comment: As submitted to journal. This version also adds a proof of fixed-parameter tractability for parameter k+t using the technique of randomized contraction

    On the Threshold of Intractability

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    We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in 2O(klogk)+poly(n)2^{O(\surd k \log k)} + \text{poly}(n) time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems

    On the (non-)existence of polynomial kernels for Pl-free edge modification problems

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    Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E \vartriangle F) satisfies the property P. In the P edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free and Cl-free edge-deletion problems for large enough l

    Contraction blockers for graphs with forbidden induced paths.

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    We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pℓ-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs

    Hitting forbidden minors: Approximation and Kernelization

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    We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most kk vertices can be deleted from a graph GG such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding tt-claw K1,tK_{1,t}, the star with tt leves, as an induced subgraph, where tt is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of O(log3/2OPT)O(\log^{3/2} OPT), where OPTOPT is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph θc\theta_c as a minor for a fixed integer cc. The graph θc\theta_c consists of two vertices connected by cc parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes
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