Abstract
We reduce the approximation factor for Vertex Cover to \(2 - \theta(\frac{1}{\sqrt{{\rm log} n}})\) (instead of the previous \(2- \theta(\frac{{\rm log log} n}{{\rm log}\ n})\), obtained by Bar-Yehuda and Even [3], and by Monien and Speckenmeyer[11]). The improvement of the vanishing factor comes as an application of the recent results of Arora, Rao, and Vazirani [2] 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 in [2]. 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 [2] translates into the existence of a big independent set.
A preliminary version of this work appeared as a McMaster University Technical Report CAS-04-05-GK and ECCC Report TR04-084, September/October 2004.
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Karakostas, G. (2005). A Better Approximation Ratio for the Vertex Cover Problem. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_84
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DOI: https://doi.org/10.1007/11523468_84
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