Abstract:
We reduce the approximation factor for Vertex Cover to $2-\Theta(1/\sqrt{logn})$ (instead of the previous $2-\Theta(loglogn/logn})$, obtained by Bar-Yehuda and Even, and by Monien and Speckenmeyer in 1985. The improvement of the vanishing factor comes as an application of the recent results of Arora, Rao, and Vazirani that improved the approximation factor of the sparsest cut and balanced cut problems. In particular, we use the existence of two big and well-separated sets of nodes in the solution of the semidefinite relaxation for balanced cut, proven by Arora et al. We observe that a solution of the semidefinite relaxation for vertex cover, when strengthened with the triangle inequalities, can be transformed into a solution of a balanced cut problem, and therefore the existence of big well-separated sets in the sense of Arora et al. translates into the existence of a big independent set.