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Improved Approximation Algorithms for Maximum Graph Partitioning Problems

Author

Listed:
  • Gerold Jäger

    (Christian-Albrechts-Universität Kiel, Mathematisches Seminar)

  • Anand Srivastav

    (Christian-Albrechts-Universität Kiel, Mathematisches Seminar)

Abstract

We consider the design of approximation algorithms for a number of maximum graph partitioning problems, among others MAX-k-CUT, MAX-k-DENSE-SUBGRAPH, and MAX-k-DIRECTED-UNCUT. We present a new version of the semidefnite relaxation scheme along with a better analysis, extending work of Halperin and Zwick (2002). This leads to an improvement over known approximation factors for such problems. The key to the improvement is the following new technique: It was already observed by Han et al. (2002) that a parameter-driven choice of the random hyperplane can lead to better approximation factors than obtained by Goemans and Williamson (1995). But it remained difficult to find a “good” set of parameters. In this paper, we analyze random hyperplanes depending on several new parameters. We prove that a sub-optimal choice of these parameters can be obtained by the solution of a linear program which leads to the desired improvement of the approximation factors. In this fashion a more systematic analysis of the semidefinite relaxation scheme is obtained.

Suggested Citation

  • Gerold Jäger & Anand Srivastav, 2005. "Improved Approximation Algorithms for Maximum Graph Partitioning Problems," Journal of Combinatorial Optimization, Springer, vol. 10(2), pages 133-167, September.
    • Handle: RePEc:spr:jcomop:v:10:y:2005:i:2:d:10.1007_s10878-005-2269-7
    DOI: 10.1007/s10878-005-2269-7
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    Citations

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    Cited by:

    1. Alain Billionnet & Sourour Elloumi & Amélie Lambert & Angelika Wiegele, 2017. "Using a Conic Bundle Method to Accelerate Both Phases of a Quadratic Convex Reformulation," INFORMS Journal on Computing, INFORMS, vol. 29(2), pages 318-331, May.
    2. Bourgeois, Nicolas & Giannakos, Aristotelis & Lucarelli, Giorgio & Milis, Ioannis & Paschos, Vangelis Th., 2017. "Exact and superpolynomial approximation algorithms for the densest k-subgraph problem," European Journal of Operational Research, Elsevier, vol. 262(3), pages 894-903.
    3. Mourad El Ouali & Helena Fohlin & Anand Srivastav, 2016. "An approximation algorithm for the partial vertex cover problem in hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 846-864, February.
    4. Federico Della Croce & Vangelis Th. Paschos, 2014. "Efficient algorithms for the max $$k$$ -vertex cover problem," Journal of Combinatorial Optimization, Springer, vol. 28(3), pages 674-691, October.

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