# Approximating the independence number via the*ϑ*-function

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## Abstract

We describe an approximation algorithm for the independence number of a graph. If a graph on*n* vertices has an independence number*n/k + m* for some fixed integer*k* ⩾ 3 and some*m* > 0, the algorithm finds, in random polynomial time, an independent set of size\(\tilde \Omega (m^{{3 \mathord{\left/ {\vphantom {3 {(k + 1)}}} \right. \kern-\nulldelimiterspace} {(k + 1)}}} )\), improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size Ω(*m*^{1/(k−1)}) in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovász*ϑ*-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the*ϑ*-function of an*n* vertex graph is at least*Mn*^{1−2/k} for some absolute constant*M*, we describe another, related, efficient algorithm that finds an independent set of size*k.* Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the*ϑ*-function and the independence number of a graph on*n* vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

### Keywords

Independence number of a graph Semi-definite programming Approximation algorithms## Preview

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