A (2 − ε)-approximation ratio for vertex cover problem on special graphs

The vertex cover problem is a famous combinatorial problem, and its complexity has been heavily studied. It is known that it is hard to approximate to within any constant factor better than 2, while a 2-approximation for it can be trivially obtained. In this paper, new properties and new techniques are introduced which lead to approximation ratios smaller than 2 on special graphs; e.g. graphs for which their maximum cut values are less than 0.999|E|. In fact, we produce a (1.9997)-approximation ratio on graph G, where the (0.878)-approximation algorithm of the Goemans-Williamson for the maximum cut problem on G produces a value smaller than 0.877122|E|.