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

TL;DR

This paper proves that the Vertex Cover problem is fixed-parameter tractable when parameterized above a new, stricter guarantee involving maximum matching and LP relaxation, with an algorithm running in $O^*(3^k)$ time.

Contribution

The paper introduces a novel above-guarantee parameterization for Vertex Cover and demonstrates fixed-parameter tractability with a new algorithm.

Findings

Vertex Cover is FPT for the above-guarantee parameterization.

Developed an $O^*(3^k)$ time algorithm for the problem.

Extends the understanding of parameterized complexity for Vertex Cover.

Abstract

We investigate the following above-guarantee parameterization of the classical Vertex Cover problem: Given a graph $G$ and $k\in\mathbb{N}$ as input, does $G$ have a vertex cover of size at most $(2LP-MM)+k$? Here $MM$ is the size of a maximum matching of $G$, $LP$ is the value of an optimum solution to the relaxed (standard) LP for Vertex Cover on $G$, and $k$ is the parameter. Since $(2LP-MM)\geq{LP}\geq{MM}$, this is a stricter parameterization than those---namely, above-$MM$, and above-$LP$---which have been studied so far. We prove that Vertex Cover is fixed-parameter tractable for this stricter parameter $k$: We derive an algorithm which solves Vertex Cover in time $O^{*}(3^{k})$, pushing the envelope further on the parameterized tractability of Vertex Cover.