A polynomial time $\frac 3 2$ -approximation algorithm for the vertex cover problem on a class of graphs

TL;DR

This paper presents a polynomial time algorithm achieving a 3/2-approximation for the vertex cover problem on certain graph classes, with extended guarantees and optimal solutions on structured graphs.

Contribution

It introduces a new 3/2-approximation algorithm for vertex cover on graphs with the active edge hypothesis and extends it to arbitrary graphs with bounded error.

Findings

Achieved a 3/2-approximation ratio on specific graph classes.

Extended the algorithm to arbitrary graphs with an error bound /2|S*|+b.

Observed zero error (b=0) on tested instances, including hard cases.

Abstract

We develop a polynomial time 3/2-approximation algorithm to solve the vertex cover problem on a class of graphs satisfying a property called ``active edge hypothesis''. The algorithm also guarantees an optimal solution on specially structured graphs. Further, we give an extended algorithm which guarantees a vertex cover $S_1$ on an arbitrary graph such that $|S_1|\leq {3/2} |S^*|+\xi$ where $S^*$ is an optimal vertex cover and $\xi$ is an error bound identified by the algorithm. We obtained $\xi = 0$ for all the test problems we have considered which include specially constructed instances that were expected to be hard. So far we could not construct a graph that gives $\xi \not= 0$.