Rank Vertex Cover as a Natural Problem for Algebraic Compression

Meesum, Syed Mohammad; Panolan, Fahad; Saurabh, Saket; Zehavi, Meirav

doi:10.1137/17m1154370

SIAM J. Discrete Math.

2019

DOI: 10.1137/17m1154370

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Rank Vertex Cover as a Natural Problem for Algebraic Compression

Syed Mohammad Meesum¹,

Fahad Panolan²,

Saket Saurabh³

et al.

Abstract: The question of the existence of a polynomial kernelization of the Vertex Cover Above LP problem has been a longstanding, notorious open problem in Parameterized Complexity. Five years ago, the breakthrough work by Kratsch and Wahlström on representative sets has finally answered this question in the affirmative [FOCS 2012].In this paper, we present an alternative, algebraic compression of the Vertex Cover Above LP problem into the Rank Vertex Cover problem. Here, the input consists of a graph G, a parameter k… Show more

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2020

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Representative Sets and Irrelevant Vertices

Kratsch¹,

Wahlström

2020

J. ACM

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We continue the development of matroid-based techniques for kernelization, initiated by the present authors [47]. We significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to Lovász [51] and Marx [53]. As a first result, we show how representative sets can be used to derive a polynomial kernel for the elusive ALMOST 2- SAT problem (where the task is to remove at most k clauses to make a 2- CNF formula satisfiable), solving a major open problem in kernelization. This result also yields a new O(√log OPT)-approximation for the problem, improving on the O(√log n)-approximation of Agarwal et al. [3] and an implicit O(log OPT)-approximation due to Even et al. [24]. We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, that is, vertices that can be made undeletable without affecting the answer to the problem. This gives the first significant progress towards a polynomial kernel for the MULTIWAY CUT problem; in particular, we get a kernel of O( k s+1 ) vertices for MULTIWAY CUT instances with at most s terminals. Both these kernelization results have significant spin-off effects, producing the first polynomial kernels for a range of related problems. More generally, the irrelevant vertex results have implications for covering min cuts in graphs. For a directed graph G=(V,E) and sets S, T ⊆ V , let r be the size of a minimum ( S,T )-vertex cut (which may intersect S and T ). We can find a set Z ⊆ V of size O(|S| . |T| . r) that contains a minimum ( A,B )-vertex cut for every A ⊆ S , B ⊆ T . Similarly, for an undirected graph G=(V,E) , a set of terminals X ⊆ V , and a constant s , we can find a set Z ⊆ V of size O(|X| s+1 ) that contains a minimum multiway cut for every partition of X into at most s pairwise disjoint subsets. Both results are polynomial time. We expect this to have further applications; in particular, we get direct, reduction rule-based kernelizations for all problems above, in contrast to the indirect compression-based kernel previously given for ODD CYCLE TRANSVERSAL [47]. All our results are randomized, with failure probabilities that can be made exponentially small in n , due to needing a representation of a matroid to apply the representative sets tool.

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Representative Sets and Irrelevant Vertices

Kratsch¹,

Wahlström

2020

J. ACM

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Rank Vertex Cover as a Natural Problem for Algebraic Compression

Cited by 1 publication

References 28 publications

Representative Sets and Irrelevant Vertices

Representative Sets and Irrelevant Vertices

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