Vertex Covers Revisited: Indirect Certificates and FPT Algorithms

TL;DR

This paper introduces new simple fixed-parameter tractable algorithms for Vertex Cover by exploiting structural properties and indirect certificates, improving understanding and solution techniques for this classic NP-complete problem.

Contribution

The paper presents six novel FPT algorithms for Vertex Cover, including the concept of indirect certificates with size at most one-third of the cover, using various algorithmic techniques.

Findings

Existence of an indirect certificate with at most k/3 vertices for graphs with a k-vertex cover

Development of three new FPT algorithms based on random partition and selection

Enhanced understanding of structural properties of vertex covers in graphs

Abstract

The classical NP-complete problem Vertex Cover requires us to determine whether a graph contains at most $k$ vertices that cover all edges. In spite of its intractability, the problem can be solved in FPT time for parameter $k$ by various techniques. In this paper, we present half a dozen new and simple FPT algorithms for Vertex Cover with respect to parameter $k$. For this purpose, we explore structural properties of vertex covers and use these properties to obtain FPT algorithms by iterative compression, colour coding, and indirect certificating methods. In particular, we show that every graph with a $k$-vertex cover admits an indirect certificate with at most $k/3$ vertices, which lays the foundation of three new FPT algorithms based on random partition and random selection.