Divide-and-Conquer Approximation Algorithm for Vertex Cover

The vertex cover problem is a classical NP-complete problem for which the best worstcase 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 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 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 n 2 on the optimal vertex cover, and are unlikely to have a large bipartite subgraph.