New insight into introducing a (2-ℇ)-approximation ratio for minimum vertex cover problem

Vertex cover problem is a famous combinatorial problem, which 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. Then, by introducing a modified graph and corresponding model along with satisfying the proposed assumptions, we propose new insight into solving this open problem and we introduce an approximation algorithm with a performance ratio of 1.999999 on arbitrary graphs.