Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

TL;DR

This paper explores new fixed-parameter tractable algorithms and lossy kernelizations for the Connected Vertex Cover problem, focusing on parameters related to graph structure rather than just solution size.

Contribution

It introduces lossy kernels and FPT algorithms based on natural graph parameters like split, clique, and cluster deletion sets, expanding beyond traditional solution size parameters.

Findings

Developed lossy kernels for various structural parameters.

Designed FPT algorithms for parameters smaller or more natural than solution size.

Extended the applicability of kernelization techniques to broader graph parameters.

Abstract

The CONNECTED VERTEX COVER problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al.[STOC 2017], have shown an $\alpha$-approximate kernel for the problem for every $\alpha > 1$, in the framework of approximate or lossy kernelization. In this work, we exhibit lossy kernels and FPT algorithms for CONNECTED VERTEX COVER for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. The parameters we consider are the sizes of a split deletion set, clique deletion set, clique cover, cluster deletion set and chordal deletion set.