A ($2-\varepsilon$)-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 $0.877122|E|$‎.