Abstract
In the literature, packing and partitioning orbitopes were discussed to handle symmetries that act on variable matrices in certain binary programs. In this paper, we extend this concept by restrictions on the number of 1-entries in each column. We develop extended formulations of the resulting polytopes and present numerical results that show their effect on the LP relaxation of a graph partitioning problem.
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References
Balas, E. (1998). Disjunctive programming: Properties of the convex hull of feasible points. Discrete Applied Mathematics, 89(1), 3–44.
Davis, T. A., & Hu, Y. (2011). The University of Florida sparse matrix collection. ACM Transactions on Mathematical Software, 38(1), 1–25.
Faenza, Y., & Kaibel, V. (2009). Extended formulations for packing and partitioning orbitopes. Mathematics of Operations Research, 34(3), 686–697.
Gamrath, G., Fischer, T., Gally, T., Gleixner, A. M., Hendel, G., Koch, T., Maher, S.J., Miltenberger, M., Müller, B., Pfetsch, M. E., Puchert, C., Rehfeldt, D., Schenker, S., Schwarz, R., Serrano, F., Shinano, Y., Vigerske, S., Weninger, D., Winkler, M., Witt, J. T., Witzig, J. (2016). The SCIP Optimization Suite 3.2. Technical Report 15-60, ZIB, Takustr. 7, 14195 Berlin.
Gawrilow, E., & Joswig, M. (2000). Polymake: A framework for analyzing convex polytopes. In Polytopes – combinatorics and computation (pp. 43–74).
Hojny, C., & Pfetsch, M. E. (2017). Polytopes associated with symmetry handling. www.optimization-online.org/DB_HTML/2017/01/5835.html.
Kaibel, V., & Loos, A. (2011). Finding descriptions of polytopes via extended formulations and liftings. In: A. R. Mahjoub (Ed.), Progress in combinatorial optimization. New Jersey: Wiley.
Kaibel, V., & Pfetsch, M. E. (2008). Packing and partitioning orbitopes. Mathematical Programming, 114(1), 1–36.
Karisch, S. E., & Rendl, F. (1998). Semidefinite programming and graph equipartition. Topics in Semidefinite and Interior-Point Methods, 18, 77–95.
Karypis, G., & Kumar, V. (1998). A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1), 359–392.
Lisser, A., & Rendl, F. (2003). Graph partitioning using linear and semidefinite programming. Mathematical Programming, 95(1), 91–101.
Loos, A. (2011). Describing orbitopes by linear inequalities and projection based tools. Ph.D. thesis, Universität Magdeburg.
Acknowledgements
The research of the second and third author was supported by the German Research Foundation (DFG) as part of the Collaborative Research Centre 666 and 805, respectively.
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Hojny, C., Pfetsch, M.E., Schmitt, A. (2018). Extended Formulations for Column Constrained Orbitopes. In: Kliewer, N., Ehmke, J., Borndörfer, R. (eds) Operations Research Proceedings 2017. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-89920-6_28
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DOI: https://doi.org/10.1007/978-3-319-89920-6_28
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