On Multiway Cut Parameterized above Lower Bounds

Cygan, Marek; Pilipczuk, Marcin; Pilipczuk, Michał; Wojtaszczyk, Jakub Onufry

doi:10.1007/978-3-642-28050-4_1

Cited by 57 publications

(85 citation statements)

References 20 publications

(45 reference statements)

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“…We also develop new rules, called the packing branching and packing reduction rules, which are inspired by these previous studies. Then, we rameter; however, many interesting results have already been obtained [6,14,11,20]. While the exact exponential algorithms do not use any lower bounds to prune the search, with this parameterization, the current fastest algorithms are based on the branch-and-bound method and use a (simple) LP lower bound to prune the search.…”

Section: Introductionmentioning

confidence: 99%

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Akiba

Iwata

2016

Theoretical Computer Science

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We investigate the gap between theory and practice for exact branching algorithms. In theory, branch-and-reduce algorithms currently have the best time complexity for numerous important problems. On the other hand, in practice, state-of-the-art methods are based on different approaches, and the empirical efficiency of such theoretical algorithms have seldom been investigated probably because they are seemingly inefficient because of the plethora of complex reduction rules. In this paper, we design a branch-and-reduce algorithm for the vertex cover problem using the techniques developed for theoretical algorithms and compare its practical performance with other state-of-the-art empirical methods. The results indicate that branch-and-reduce algorithms are actually quite practical and competitive with other state-ofthe-art approaches for several kinds of instances, thus showing the practical impact of theoretical research on branching algorithms.

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Section: Introductionmentioning

confidence: 99%

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Akiba

Iwata

2016

Theoretical Computer Science

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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%

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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“…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]. Last but not least, the research on the OCT problem resulted in the introduction of iterative compression, a simple yet powerful technique for designing parameterized algorithms [25].…”

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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On Multiway Cut Parameterized above Lower Bounds

Cited by 57 publications

References 20 publications

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Parameterizing Edge Modification Problems Above Lower Bounds

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

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