p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms

Carl Einarson; Gregory Gutin; Bart M. P. Jansen; Diptapriyo Majumdar; Magnus Wahlstrom

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arXiv · 2009.08158

p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms

Carl Einarson·Gregory Gutin·Bart M. P. Jansen·Diptapriyo Majumdar·Magnus Wahlstrom

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Abstract

We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP \subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).

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Carl Einarson, Gregory Gutin, Bart M. P. Jansen, Diptapriyo Majumdar, Magnus Wahlstrom. 2020-09-17. p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms. https://arxiv.org/abs/2009.08158

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