Divide-and-Conquer Approximation Algorithm for Vertex Cover

Abstract

The vertex cover problem is a classical NP-complete problem for which the best worst-case approximation ratio is $2-o(1)$. In this paper, we use a collection of simple graph transformations, each of which guarantees an approximation ratio of $\frac{3}{2}$, to find approximate vertex covers for a large collection of randomly generated graphs and test graphs from various sources. The graph reductions are extremely fast, and even though they by themselves are not guaranteed to find a vertex cover, we manage to find a $\frac{3}{2}$-approximate vertex cover for almost every single graph in our collection. The few graphs that we cannot solve have specific structure: they are triangle-free, with a minimum degree of at least 3, a lower bound of $\frac{n}{2}$ on the optimal vertex cover, and are unlikely to have a large bipartite subgraph.