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.
Financial support from the German Science Foundation is acknowledged under contracts Bu 2313/1-1 and Li 1675/1-1. Partially supported by the Marie Curie RTN Adonet 504438 funded by the EU.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
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)
Barahona, F., Mahjoub, A.R.: On the cut polytope. Mathematical Programming 36, 157–173 (1986)
Buchheim, C., Liers, F., Oswald, M.: Local cuts revisited. Operations Research Letters (2008), doi:10.1016/j.orl.2008.01.004
De Simone, C.: The cut polytope and the Boolean quadric polytope. Discrete Mathematics 79, 71–75 (1990)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol. 15. Springer, Heidelberg (1997)
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)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Buchheim, C., Liers, F., Oswald, M. (2008). A Basic Toolbox for Constrained Quadratic 0/1 Optimization. In: McGeoch, C.C. (eds) Experimental Algorithms. WEA 2008. Lecture Notes in Computer Science, vol 5038. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68552-4_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-68552-4_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68548-7
Online ISBN: 978-3-540-68552-4
eBook Packages: Computer ScienceComputer Science (R0)