Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms 2015
DOI: 10.1137/1.9781611974331.ch80
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Raising The Bar For Vertex Cover: Fixed-parameter Tractability Above A Higher Guarantee

Abstract: The standard parameterization of the Vertex Cover problem (Given an undirected graph G and k ∈ N as input, does G have a vertex cover of size at most k?) has the solution size k as the parameter. The following more challenging parameterization of Vertex Cover stems from the observation that the size MM of a maximum matching of G lower-bounds the size of any vertex cover of G: Does G have a vertex cover of size at most MM + kμ? The parameter is the excess kμ of the solution size over the lower bound MM.Razgon … Show more

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Cited by 32 publications
(35 citation statements)
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“…A more desirable and stronger result would be a constant-factor approximation of the difference d between the optimal solution cost and a lower bound or a fixed-parameter tractability result with respect to the parameter d [6,11,17,24]. However, we show that such algorithms (presumably) do not exist.…”
Section: Parameterized Hardness and Inapproximabilitymentioning
confidence: 85%
“…A more desirable and stronger result would be a constant-factor approximation of the difference d between the optimal solution cost and a lower bound or a fixed-parameter tractability result with respect to the parameter d [6,11,17,24]. However, we show that such algorithms (presumably) do not exist.…”
Section: Parameterized Hardness and Inapproximabilitymentioning
confidence: 85%
“…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%
“…Two well-known lower bounds for the size of vertex covers for a graph G = (V, E) are the maximum size of a matching of G and the smallest size of fractional vertex covers for G; we (essentially) follow Garg and Philip [11] in denoting these two values by M M (G) and LP (G). Note that the notation LP (G) comes from the fact that fractional vertex covers come up naturally in the linear programming relaxation of the vertex cover problem, where we must assign each vertex a fractional value such that each edge is incident with total value of at least 1.…”
Section: Introductionmentioning
confidence: 99%
“…Recently, Garg and Philip [11] made an important contribution to understanding the parameterized complexity of the vertex cover problem by proving it to be FPT with respect to parameter ℓ = k−(2LP (G)−M M (G)). Building on an observation of Lovász and Plummer [20] they prove that V C(G) ≥ 2LP (G) − M M (G), i.e., that 2LP (G) − M M (G) is indeed a lower bound for the minimum vertex covers size of any graph G. They then design a branching algorithm with running time O * (3 ℓ ) that builds on the well-known Gallai-Edmonds decomposition for maximum matchings to guide its branching choices.…”
Section: Introductionmentioning
confidence: 99%