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Approximating the Independence Number and the Chromatic Number in Expected Polynomial Time

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Automata, Languages and Programming (ICALP 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

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Abstract

The independence number of a graph and its chromatic number are hard to approximate. It is known that, unless coRP = NP, there is no polynomial time algorithm which approximates any of these quantities within a factor of n1-ε for graphs on n vertices.

We show that the situation is significantly better for the average case. For every edge probability p = p(n) in the range n −1/2+ε ≤ p ≤ 3/4, we present an approximation algorithm for the independence number of graphs on n vertices, whose approximation ratio is O((np)1/2/ log n) and whose expected running time over the probability space G(n, p) is polynomial. An algorithm with similar features is described also for the chromatic number.

A key ingredient in the analysis of both algorithms is a new large deviation inequality for eigenvalues of random matrices, obtained through an application of Talagrand’s inequality.

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Author information

Authors and Affiliations

  1. Department of Mathematics Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Michael Krivelevich

  2. Microsoft Research, 1 Microsoft Way, Redmond, WA, 98052, USA

    H. Van Vu

Authors
  1. Michael Krivelevich
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  2. H. Van Vu
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Editor information

Editors and Affiliations

  1. Department of Computer Sciences, University of Pisa, Corso Italia, 40, 56125, Pisa, Italy

    Ugo Montanari

  2. Center for Computer Sciences, University of Geneva, 24, Rue Général Dufour, 1211, Geneva 4, Switzerland

    José D. P. Rolim

  3. Department of Computer Sciences, ETH Zurich, 8092, Zurich, Switzerland

    Emo Welzl

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© 2000 Springer-Verlag Berlin Heidelberg

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Krivelevich, M., Van Vu, H. (2000). Approximating the Independence Number and the Chromatic Number in Expected Polynomial Time. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_3

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  • DOI: https://doi.org/10.1007/3-540-45022-X_3

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

  • eBook Packages: Springer Book Archive

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