Approximating the independence number via the theta-function

Noga Alon, Nabil Kahale

Abstract: We describe an approximation algorithm for the independence number of a graph. If a graph onn vertices has an independence numbern/k + m for some fixed integerk ⩾ 3 and somem > 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 Lovászϑ-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If theϑ-function of ann vertex graph is at leastMn 1−2/k for some absolute constantM, we describe another, related, efficient algorithm that finds an independent set of sizek. 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 theϑ-function and the independence number of a graph onn vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.