A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

Kratsch, Stefan

doi:10.1137/16m1104585

Cited by 14 publications

(11 citation statements)

References 26 publications

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“…Kratsch and Wahlström [28] gave the first (randomized) polynomial kernelization for Vertex Cover parameterized above the optimum value of the standard LP relaxation. This result was later strengthened by Kratsch [27] who parameterized above an even stronger lower bound on the solution, 2LP − MM, where LP denotes the optimum value of the standard LP relaxation and MM denotes the size of the maximum matching in the input graph. Majumdar et al [33] considered as their parameter the deletion distance to graphs of degree 2 and to cluster graphs where each clique has bounded size.…”

Section: Our Results and Significance Of The Chosen Parameterizationsmentioning

confidence: 99%

On the Approximate Compressibility of Connected Vertex Cover

2020

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The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have a polynomial kernelization algorithm. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokshtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0.In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs.

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Section: Our Results and Significance Of The Chosen Parameterizationsmentioning

confidence: 99%

On the Approximate Compressibility of Connected Vertex Cover

2020

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“…It has also been shown that Vertex Cover is FPT for even smaller above guarantee parameters such as k − ℓ [6,28] and k − 2ℓ + m [17], where ℓ is the optimal LP relaxation value of Vertex Cover. Kernelization with respect to these parameters has also been studied [25,26]. This work considers above guarantee parameterizations of Vertex Cover where the lower bounds are structural parameters not related to the matching number, such as feedback vertex number, degeneracy, and cluster vertex deletion number.…”

Section: Introductionmentioning

confidence: 99%

Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

Kellerhals¹,

Koana²,

Kunz³

2022

Preprint

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Vertex Cover parameterized by the solution size k is the quintessential fixedparameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.

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“…The kernelization complexity of the solution-size parameterization of F-Deletion has been the subject of intensive research [17,18,22,28,40]. In this work we attempt to find the widest class of structural parameterizations for which F-Deletion admits polynomial kernels, continuing a long line of investigation into structural parameterizations for Vertex Cover [4,19,25,30,31,33], Feedback Vertex Set [27,32], and other F-Deletion problems [16,21].…”

Section: Introductionmentioning

confidence: 99%

Polynomial kernels for hitting forbidden minors under structural parameterizations

Jansen

Pieterse

2020

Theoretical Computer Science

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We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of graphs F, the F-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when F contains a planar graph, an instance (G, k) can be reduced in polynomial time to an equivalent one of size k O(1). In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the F-Deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth η. We prove that for each set F of connected graphs and constant η, the F-Deletion problem parameterized by the size of a treedepth-η modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on F-Deletion. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G − X. Using the language of labeled minors, we analyze the fragments of potential forbidden minor models that can remain after removing an optimal F-Deletion solution from a single connected component of G − X. By bounding the number of different types of behavior that can occur by a polynomial in |X|, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of F-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes earlier preprocessing results for F-Deletion parameterized by a vertex cover, which is a treedepth-one modulator.

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A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

Cited by 14 publications

References 26 publications

On the Approximate Compressibility of Connected Vertex Cover

On the Approximate Compressibility of Connected Vertex Cover

Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

Polynomial kernels for hitting forbidden minors under structural parameterizations

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