Abstract. We reduce the approximation factor for the vertex cover to 2 − ( 1 √ log n ) (instead of the previous 2 − ln ln n 2ln n obtained by Bar-Yehuda and Even [1985] and Monien and Speckenmeyer [1985] ). The improvement of the vanishing factor comes as an application of the recent results of Arora et al. [2004] 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. [2004]. 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. [2004] translates into the existence of a big independent set.
These were finally improved by Karakostas REF who used the ARV Structural Theorem to provide a (2 − Ω(1/ log n))-approximation for the problem.
A better approximation ratio for the vertex cover problem
TALG
Mathematics Computer Science