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

TL;DR

This paper introduces and analyzes parameterized and approximation algorithms for two generalizations of the Connected Vertex Cover problem, focusing on their fixed-parameter tractability, kernelization limits, and approximation ratios.

Contribution

It extends the study of Connected Vertex Cover to p-Edge-Connected and p-Vertex-Connected variants, providing new algorithms and kernelization results.

Findings

Both problems are fixed-parameter tractable.

They do not admit polynomial kernels unless NP ⊆ coNP/poly.

Existence of polynomial-sized approximate kernelization schemes.

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).