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An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2006, RANDOM 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4110))

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Abstract

Given a positive integer p and a complete graph with non-negative edge weights that satisfy the triangle inequality, the remote-clique problem is to find a subset of p vertices having a maximum-weight induced subgraph. A greedy algorithm for the problem has been shown to have an approximation ratio of 4, but this analysis was not shown to be tight. In this paper, we present an algorithm called d-Greedy Augment that generalizes this greedy algorithm (they are equivalent when d = 1). We use the technique of factor-revealing linear programs to prove that d-Greedy Augment, which has a running time of O(pdn d), achieves an approximation ratio of (2p – 2)/(p + d – 2). Thus, when d = 1, d-Greedy Augment achieves an approximation ratio of 2 and runs in time O(pn), making it the fastest known 2-approximation for the remote-clique problem. The usefulness of factor-revealing LPs in the analysis of d-Greedy Augment suggests possible applicability of this technique to the study of other approximation algorithms.

This research was supported in part by the National Science Foundation under grant 0305954. It was performed while the first author was at Washington University in St. Louis.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Washington, Seattle, Seattle, WA, 98195, USA

    Benjamin E. Birnbaum

  2. Department of Computer Science and Engineering, Washington University in St. Louis, St. Louis, MO, 63130, USA

    Kenneth J. Goldman

Authors
  1. Benjamin E. Birnbaum
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  2. Kenneth J. Goldman
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord - Ed. Omega, 240 Jordi Girona Salgado, 1-3 E-08034, Barcelona

    Josep Díaz

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

    Klaus Jansen

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

    José D. P. Rolim

  4. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Uri Zwick

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© 2006 Springer-Verlag Berlin Heidelberg

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Birnbaum, B.E., Goldman, K.J. (2006). An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 2006. Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_7

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  • DOI: https://doi.org/10.1007/11830924_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38044-3

  • Online ISBN: 978-3-540-38045-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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