Mathematical Programming

, Volume 80, Issue 3, pp 253–264

# Approximating the independence number via theϑ-function

• Noga Alon
• Nabil Kahale
Article

## 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(m1/(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 leastMn1−2/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.

### Keywords

Independence number of a graph Semi-definite programming Approximation algorithms

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© The Mathematical Programming Society, Inc. 1998

## Authors and Affiliations

• Noga Alon
• 1
• 2
• Nabil Kahale
• 2
1. 1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
2. 2.AT & T Bell LaboratoriesMurray HillUSA