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TR04-084 | 28th September 2004 00:00
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#### A better approximation ratio for the Vertex Cover problem

**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.