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An Improved Decomposition Theorem for Graphs Excluding a Fixed Minor

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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

Given a graph G and a parameter δ, we want to decompose the graph into clusters of diameter δ without cutting too many edges. For any graph that excludes a K r,r minor, Klein, Plotkin and Rao [15] showed that this can be done while cutting only O(r 3/δ) fraction of the edges. This implies a bound on multicommodity max-flow min-cut ratio for such graphs. This result as well as the decomposition theorem have found numerous applications to approximation algorithms and metric embeddings for such graphs.

In this paper, we improve the above decomposition results from O(r 3) to O(r 2). This shows that for graphs excluding any minor of size r, the multicommodity max-flow min-cut ratio is at most O(r 2) (for the uniform demand case). This also improves the performance guarantees of several applications of the decomposition theorem.

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Author information

Authors and Affiliations

  1. Kasetsart University, Bangkok, Thailand

    Jittat Fakcharoenphol

  2. Computer Science Division, University of California, Berkeley

    Kunal Talwar

Authors
  1. Jittat Fakcharoenphol
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  2. Kunal Talwar
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Editor information

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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Fakcharoenphol, J., Talwar, K. (2003). An Improved Decomposition Theorem for Graphs Excluding a Fixed Minor. 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_4

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  • DOI: https://doi.org/10.1007/978-3-540-45198-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40770-6

  • Online ISBN: 978-3-540-45198-3

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