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A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations

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Integer Programming and Combinatorial Optimization (IPCO 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4513))

Abstract

In this paper we present a method for finding exact solutions of the Max-Cut problem max x T Lx such that x ∈ { − 1,1}n. We use a semidefinite relaxation combined with triangle inequalities, which we solve with the bundle method. This approach is due to [12] and uses Lagrangian duality to get upper bounds with reasonable computational effort. The expensive part of our bounding procedure is solving the basic semidefinite programming relaxation of the Max-Cut problem.

We review other solution approaches and compare the numerical results with our method. We also extend our experiments to unconstrained quadratic 0-1 problems and to instances of the graph bisection problem.

The experiments show, that our method nearly always outperforms all other approaches. Our algorithm, which is publicly accessible through the Internet, can solve virtually any instance with about 100 variables in a routine way.

Supported in part by the EU project Algorithmic Discrete Optimization (ADONET), MRTN-CT-2003-504438.

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Authors and Affiliations

  1. Alpen-Adria-Universität Klagenfurt, Institut für Mathematik, Universitätsstr. 65-67, 9020 Klagenfurt, Austria

    Franz Rendl & Angelika Wiegele

  2. Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” – CNR, Viale Manzoni, 30, 00185 Roma, Italy

    Giovanni Rinaldi

Authors
  1. Franz Rendl
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  2. Giovanni Rinaldi
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  3. Angelika Wiegele
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Matteo Fischetti David P. Williamson

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Rendl, F., Rinaldi, G., Wiegele, A. (2007). A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_23

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  • DOI: https://doi.org/10.1007/978-3-540-72792-7_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72791-0

  • Online ISBN: 978-3-540-72792-7

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