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A Basic Toolbox for Constrained Quadratic 0/1 Optimization

  • Christoph Buchheim
  • Frauke Liers
  • Marcus Oswald
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5038)

Abstract

In many practical applications, the task is to optimize a non-linear function over a well-studied polytope P as, e.g., the matching polytope or the travelling salesman polytope (TSP). In this paper, we focus on quadratic objective functions. Prominent examples are the quadratic assignment and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, they have to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. By extensive experiments, we show that both methods can drastically accelerate the solution of constrained quadratic 0/1 problems.

Keywords

quadratic programming maximum cut problem local cuts crossing minimization similar subgraphs 

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References

  1. 1.
    Applegate, A., Bixby, R., Chvátal, V., Cook, W.: TSP cuts which do not conform to the template paradigm. In: Jünger, M., Naddef, D. (eds.) Computational Combinatorial Optimization. LNCS, vol. 2241, pp. 261–304. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Barahona, F., Mahjoub, A.R.: On the cut polytope. Mathematical Programming 36, 157–173 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Buchheim, C., Liers, F., Oswald, M.: Local cuts revisited. Operations Research Letters (2008), doi:10.1016/j.orl.2008.01.004Google Scholar
  4. 4.
    De Simone, C.: The cut polytope and the Boolean quadric polytope. Discrete Mathematics 79, 71–75 (1990)zbMATHCrossRefGoogle Scholar
  5. 5.
    Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol. 15. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  6. 6.
    Liers, F., Jünger, M., Reinelt, G., Rinaldi, G.: Computing Exact Ground States of Hard Ising Spin Glass Problems by Branch-and-Cut. New Optimization Algorithms in Physics, pp. 47–68. Wiley-VCH, Chichester (2004)Google Scholar
  7. 7.
    Rendl, F., Rinaldi, G., Wiegele, A.: A branch and bound algorithm for Max-Cut based on combining semidefinite and polyhedral relaxations. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 295–309. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christoph Buchheim
    • 1
  • Frauke Liers
    • 1
  • Marcus Oswald
    • 2
  1. 1.Institut für InformatikUniversität zu KölnKölnGermany
  2. 2.Institut für InformatikUniversität HeidelbergHeidelbergGermany

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