Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

Kratsch, Stefan; Wahlström, Magnus

doi:10.1137/1.9781611973099.8

Cited by 58 publications

(90 citation statements)

References 63 publications

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“…The study of kernelization is a major research frontier of Parameterized Complexity and many important recent advances in the area are on kernelization. These include general results showing that certain classes of parameterized problems have polynomial kernels [3,16,50,60]. The recent development of a framework for ruling out polynomial kernels under certain complexity-theoretic assumptions [15,35,51] has added a new dimension to the field and strengthened its connections to classical complexity.…”

mentioning

confidence: 99%

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Fomin

Lokshtanov²,

Misra

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

128

174

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Let F be a finite set of graphs. In the F-Deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth η-Deletion.In this paper we obtain a number of generic algorithmic results about F-Deletion, when F contains at least one planar graph. The highlights of our work are• A constant factor approximation algorithm for the optimization version of F-Deletion;• A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2 O(k) n), for the parameterized version of F-Deletion where all graphs in F are connected;• A polynomial kernel for parameterized F-Deletion.These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results -constant factor approximation, polynomial kernelization and FPT algorithms -are stringed together by a common theme of polynomial time preprocessing.

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mentioning

confidence: 99%

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Fomin

Lokshtanov²,

Misra

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

128

174

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“…The notions used in the paper of Marx have been useful in settling the parameterized complexity, as well as obtaining improved FPT algorithms for a wide variety of problems including Directed Feedback vertex Set [3], Almost 2-SAT [32] and AGVC [32,30]. More recent results on these problems have been in the area of kernelization where OCT, Almost 2-SAT and AGVC have been shown to have polynomial kernels [16,17]. These sequence of results have led to an entirely new and very active subarea dealing with parameterized graph separation problems both due to independent interest in the problems themselves, as well as due to the fact that these problems seem to be able to capture the underlying properties of a large variety of seemingly unrelated problems.…”

Section: Related Results On Cut Problemsmentioning

confidence: 99%

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

Ramanujan¹,

Saurabh²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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A skew-symmetric graph (D = (V, A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skew-symmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, d-SkewSymmetric Multicut, where we are given a skewsymmetric graph D, a family of T of d-sized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different strongly connected components of D = (V, A \ (X ∪ σ(X)). In this paper, we give an algorithm for d-Skew-Symmetric Multicut which runs in time O((4d) k (m+n+ )), where m is the number of arcs in the graph, n the number of vertices and the length of the family given in the input.This problem, apart from being independently interesting, also abstracts out and captures the main combinatorial obstacles towards solving numerous classical problems. Our algorithm for d-Skew-Symmetric Multicut paves the way for the first linear time parameterized algorithms for several problems. We demonstrate its utility by obtaining the following linear time parameterized algorithms.• We show that Almost 2-SAT is a special case of 1-Skew-Symmetric Multicut, resulting in an algorithm for Almost 2-SAT which runs in time O(4 k k 4 ) where k is the size of the solution and is the length of the input formula. Then, using linear time parameter preserving reductions to Almost 2-SAT, we obtain algorithms for Odd Cycle Transversal and Edge Bipartization which run in time O(4 k k 4 (m+n)) and O(4 k k 5 (m+ n)) respectively where k is size of the solution, * Supported by Parameterized Approximation, ERC Starting Grant 306992.

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“…The next natural question is whether Subset-DFVS has a polynomial kernel or can we rule out such a possibility under some standard assumptions? The recent developments [12,30,29] in the field of kernelization may be useful in answering this question. In the field of exact exponential algorithms, Razgon [41] gave an O * (1.9977 n ) algorithm for DFVS which was used by Chitnis et al [10] to give an O * (1.9993 n ) algorithm for the more general Subset-DFVS problem.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Chitnis

Cygan

Hajiaghayi

et al. 2015

ACM Trans. Algorithms

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Given a graph G and an integer k, the Feedback Vertex Set (FVS) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph. The first fixed-parameter algorithm for FVS in undirected graphs appeared in a monograph of Mehlhorn in 1984. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08, JACM '08) showed that it is fixed-parameter tractable by giving a 4 k k!n O(1) time algorithm. There are two subset versions of this problems: we are given an additional subset S of vertices (resp., edges) and we want to hit all cycles passing through a vertex of S (resp. an edge of S); the two variants are known to be equivalent in the parameterized sense. Recently, the Subset Feedback Vertex Set problem in undirected graphs was The technique of random sampling of important separators was used by Marx and Razgon (STOC '11, SICOMP '14) to show that Undirected Multicut is FPT and was generalized by Chitnis et al. (SODA '12, SICOMP '13) to directed graphs to show that Directed Multiway Cut is FPT. Besides proving the fixed-parameter tractability of Directed Subset Feedback Vertex Set, we reformulate the random sampling of important separators technique in an abstract way that can be used for a general family of transversal problems. We believe this general approach will be useful for showing the fixed-parameter tractability of other problems * A preliminary version of this paper appeared in ICALP 2012 [9].

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Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

Cited by 58 publications

References 63 publications

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

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