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The Lovász Number of Random Graphs

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Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques (RANDOM 2003, APPROX 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2764))

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Abstract

We study the Lovász number ϑ along with two further SDP relaxations ϑ 1/2, ϑ 2 of the independence number and the corresponding relaxations \(\bar\vartheta\), \(\bar\vartheta_{1/2}\), \(\bar\vartheta_2\) of the chromatic number on random graphs G n, p . We prove that \(\bar\vartheta,\bar\vartheta_{1/2},\bar\vartheta_2(G_{n,p})\) in the case p<n  − 1/2 −  ε are concentrated in intervals of constant length. Moreover, we estimate the probable value of \(\vartheta,\bar\vartheta(G_{n,p})\) etc. for essentially the entire range of edge probabilities p. As applications, we give improved algorithms for approximating α(G n, p ) and for deciding k-colorability in polynomial expected time.

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Authors and Affiliations

  1. Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Amin Coja-Oghlan

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  1. Amin Coja-Oghlan
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Editors and Affiliations

  1. Computer Science Department, Princeton University, 35 Olden Street, 08540, Princeton, NJ

    Sanjeev Arora

  2. Institute for Computer Science, University of Kiel, Olshausenstrasse 40, 24118, Kiel, Germany

    Klaus Jansen

  3. Battelle Bâtiment A, Centre Universitaire d’Informatique, Route de Drize 7, 1227, Carouge, Geneva, Switzerland

    José D. P. Rolim

  4. University of California, Los Angeles

    Amit Sahai

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© 2003 Springer-Verlag Berlin Heidelberg

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Coja-Oghlan, A. (2003). The Lovász Number of Random Graphs. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. RANDOM APPROX 2003 2003. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45198-3_20

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  • DOI: https://doi.org/10.1007/978-3-540-45198-3_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40770-6

  • Online ISBN: 978-3-540-45198-3

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