Raising The Bar For V<scp>ertex</scp> C<scp>over</scp>: Fixed-parameter Tractability Above A Higher Guarantee

Garg, Shivam; Philip, Geevarghese

doi:10.1137/1.9781611974331.ch80

Cited by 32 publications

(35 citation statements)

References 18 publications

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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%

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Bentert

Bevern

Nichterlein

et al. 2017

Algorithms for Sensor Systems

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We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted n-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial time if one can find a set of obligatory edges that yield a spanning subgraph with O(log n) connected components. We also provide a linear-time algorithm that reduces any input graph that consists of a tree together with g additional edges to an equivalent graph with O(g) vertices. Based on this, we obtain a polynomial-time algorithm for g ∈ O(log n). On the negative side, we show that o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard and that there are presumably no exact algorithms running in 2 o(n) time or in f (d) · n O(1) time for any computable function f .

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Section: Parameterized Hardness and Inapproximabilitymentioning

confidence: 85%

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Bentert

Bevern

Nichterlein

et al. 2017

Algorithms for Sensor Systems

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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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“…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%

A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

Kratsch¹

2018

SIAM J. Discrete Math.

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In the vertex cover problem we are given a graph G = (V, E) and an integer k and have to determine whether there is a set X ⊆ V of size at most k such that each edge in E has at least one endpoint in X. The problem can be easily solved in time O * (2 k ), making it fixed-parameter tractable (FPT) with respect to k. While the fastest known algorithm takes only time O * (1.2738 k ), much stronger improvements have been obtained by studying parameters that are smaller than k. Apart from treewidth-related results, the arguably best algorithm for vertex cover runs in time O * (2.3146 p ), where p = k − LP (G) is only the excess of the solution size k over the best fractional vertex cover (Lokshtanov et al. TALG 2014). Since p ≤ k but k cannot be bounded in terms of p alone, this strictly increases the range of tractable instances.Recently, Garg and Philip (SODA 2016) greatly contributed to understanding the parameterized complexity of the vertex cover problem. They prove that 2LP (G) − M M (G) is a lower bound for the vertex cover size of G, where M M (G) is the size of a largest matching of G, and proceed to study parameter ℓ = k − (2LP (G) − M M (G)). They give an algorithm of running time O * (3 ℓ ), proving that vertex cover is FPT in ℓ. It can be easily observed that ℓ ≤ p whereas p cannot be bounded in terms of ℓ alone. We complement the work of Garg and Philip by proving that vertex cover admits a randomized polynomial kernelization in terms of ℓ, i.e., an efficient preprocessing to size polynomial in ℓ. This improves over parameter p = k − LP (G) for which this was previously known (Kratsch and Wahlström FOCS 2012).

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Raising The Bar For Vertex Cover: Fixed-parameter Tractability Above A Higher Guarantee

Cited by 32 publications

References 18 publications

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Parameterizing Edge Modification Problems Above Lower Bounds

A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

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