Almost 2-SAT is fixed-parameter tractable

Razgon, Igor; O’Sullivan, Barry

doi:10.1016/j.jcss.2009.04.002

Cited by 107 publications

(121 citation statements)

References 21 publications

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“…The packing H and k stay unchanged, and so does . Thus, we obtain the following result: Corollary 7.8 shows that the known above-guarantee fixed-parameter algorithms for Vertex Cover [13,22,36,43] do not generalize to d-Uniform Hitting Set.…”

Section: Hard Vertex Deletion Problemsmentioning

confidence: 95%

“…The idea is to use a lower bound h on the solution size and to use := k − h as parameter instead of k. This idea has been applied successfully to Vertex Cover, the problem of finding at most k vertices such that their deletion removes all edges (that is, all K 2 s) from G. Since the size of a smallest vertex cover is large in many input graphs, parameterizations above the lower bounds "size of a maximum matching M in the input graph" and "optimum value L of the LP relaxation of the standard ILP-formulation of Vertex Cover" have been considered. After a series of improvements [13,22,36,43], the current best running time is 3 · n O (1) , where := k − (2 · L − |M|) [22].…”

Section: Introductionmentioning

confidence: 99%

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Parameterizing Edge Modification Problems Above Lower Bounds

Bevern

Froese

Komusiewicz

2016

Computer Science – Theory and Applications

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We study the parameterized complexity of a variant of the F-free Editing problem: Given a graph G and a natural number k, is it possible to modify at most k edges in G so that the resulting graph contains no induced subgraph isomorphic to F? In our variant, the input additionally contains a vertex-disjoint packing H of induced subgraphs of G, which provides a lower bound h(H) on the number of edge modifications required to transform G into an F-free graph. While earlier works used the number k as parameter or structural parameters of the input graph G, we consider instead the parameter := k − h(H), that is, the number of edge modifications above the lower bound h(H). We develop a framework of generic data reduction rules to show fixed-parameter tractability with respect to for K 3 -Free Editing, Feedback Arc Set in Tournaments, and Cluster Editing when the packing H contains subgraphs with bounded solution size. For K 3 -Free Editing, we also prove NP-hardness in case of edge-disjoint packings of K 3 s and = 0, while for K q -Free Editing and q ≥ 6, NP-hardness for = 0 even holds for vertex-disjoint packings of K q s. In addition, we provide NP-hardness results for F-free Vertex Deletion, were the aim is to delete a minimum number of vertices to make the input graph F-free.

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Section: Hard Vertex Deletion Problemsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Parameterizing Edge Modification Problems Above Lower Bounds

Bevern

Froese

Komusiewicz

2016

Computer Science – Theory and Applications

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“…This satisfiability problem belongs to the family of clause deletion problems (e.g., Almost 2-SAT [26,4,20]), where the goal is to make a formula satisfiable by the deletion of at most k clauses. …”

Section: Relation Between Dl-hom(h) and Satisfiability Problemsmentioning

confidence: 99%

List H-Coloring a Graph by Removing Few Vertices

Chitnis

Egri

Marx

2013

Lecture Notes in Computer Science

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Abstract. In the deletion version of the list homomorphism problem, we are given graphs G and H, a list L(v) ⊆ V (H) for each vertex v ∈ V (G), and an integer k. The task is to decide whether there exists a set W ⊆ V (G) of size at most k such that there is a homomorphism from G \ W to H respecting the lists. We show that DL-Hom(H), parameterized by k and |H|, is fixed-parameter tractable for any (P6, C6)-free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DLHom(H) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. [9], a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

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“…The search for a polynomial kernel for FVS lead to surprising applications of deep combinatorial results such as Gallai's theorem [26], which has also been found useful in designing FPT algorithms [10]. While investigating the graph cut problems such as MWC, Márx [21] introduced the important separator technique, which turned out to be very robust and is now the key ingredient in parameterized algorithms for various problems such as variants of FVS [5,10] or Almost 2-SAT [24]. Moreover, the recent developments on MWC show applicability of linear programming in parameterized complexity, leading to the fastest currently known algorithms not only for MWC, but also Almost 2-SAT and OCT [9,22].…”

Section: Introductionmentioning

confidence: 99%

On Group Feedback Vertex Set Parameterized by the Size of the Cutset

Cygan

Pilipczuk

2014

Algorithmica

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Copies of full items can be used for personal research or study, educational, or not-forprofit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher statement:The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-014-9966-5 A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. Abstract We study parameterized complexity of a generalization of the classical Feedback Vertex Set problem, namely the Group Feedback Vertex Set problem: we are given a graph G with edges labeled with group elements, and the goal is to compute the smallest set of vertices that hits all cycles of G that evaluate to a non-null element of the group. This problem generalizes not only Feedback Vertex Set, but also Subset Feedback Vertex Set, Multiway Cut and Odd Cycle Transversal. Completing the results of Guillemot [Discr. Opt. 2011], we provide a fixed-parameter algorithm for the parameterization by the size of the cutset only. Our algorithm works even if the group is given as a blackbox performing group operations.A preliminary version of this paper has been presented at WG 2012 [8].

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Almost 2-SAT is fixed-parameter tractable

Cited by 107 publications

References 21 publications

Parameterizing Edge Modification Problems Above Lower Bounds

Parameterizing Edge Modification Problems Above Lower Bounds

List H-Coloring a Graph by Removing Few Vertices

On Group Feedback Vertex Set Parameterized by the Size of the Cutset

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