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Orbital Independence in Symmetric Mathematical Programs

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Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

Abstract

It is well known that symmetric mathematical programs are harder to solve to global optimality using Branch-and-Bound type algorithms, since the solution symmetry is reflected in the size of the Branch-and-Bound tree. It is also well known that some of the solution symmetries are usually evident in the formulation, making it possible to attempt to deal with symmetries as a preprocessing step. One of the easiest approaches is to “break” symmetries by adjoining some symmetry-breaking constraints to the formulation, thereby removing some symmetric global optima, then solve the reformulation with a generic solver. Sets of such constraints can be generated from each orbit of the action of the symmetries on the variable index set. It is unclear, however, whether and how it is possible to choose two or more separate orbits to generate symmetry-breaking constraints which are compatible with each other (in the sense that they do not make all global optima infeasible). In this paper we discuss a new concept of orbit independence which clarifies this issue.

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References

  1. Bomze, I., Budinich, M., Pardalos, P., Pelillo, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, pp. 1–74. Kluwer Academic Publishers, Dordrecht (1998)

    Google Scholar 

  2. Costa, A., Hansen, P., Liberti, L.: Formulation symmetries in circle packing. In: Mahjoub, R. (ed.) Proceedings of the International Symposium on Combinatorial Optimization. Electronic Notes in Discrete Mathematics, vol. 36, pp. 1303–1310. Elsevier, Amsterdam (2010)

    Google Scholar 

  3. Fourer, R., Gay, D.: The AMPL Book. Duxbury Press, Pacific Grove (2002)

    Google Scholar 

  4. Galli, S.: Parsing AMPL internal format for linear and non-linear expressions (2004), B.Sc. dissertation, DEI, Politecnico di Milano, Italy

    Google Scholar 

  5. IBM: ILOG CPLEX 12.6 User’s Manual. IBM (2014)

    Google Scholar 

  6. Kaibel, V., Pfetsch, M.: Packing and partitioning orbitopes. Math. Program. 114(1), 1–36 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liberti, L.: Reformulations in mathematical programming: definitions and systematics. RAIRO-RO 43(1), 55–86 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liberti, L.: Reformulations in mathematical programming: automatic symmetry detection and exploitation. Math. Program. A 131, 273–304 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liberti, L.: Symmetry in mathematical programming. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, IMA, vol. 154, pp. 263–286. Springer, New York (2012)

    Chapter  Google Scholar 

  10. Liberti, L., Cafieri, S., Savourey, D.: The reformulation-optimization software engine. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 303–314. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  11. Liberti, L., Ostrowski, J.: Stabilizer-based symmetry breaking constraints for mathematical programs. J. Global Optim. 60, 183–194 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Margot, F.: Pruning by isomorphism in branch-and-cut. Math. Program. 94, 71–90 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Margot, F.: Exploiting orbits in symmetric ILP. Math. Program. B 98, 3–21 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Margot, F.: Symmetry in integer linear programming. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L. (eds.) 50 Years of Integer Programming, pp. 647–681. Springer, Berlin (2010)

    Google Scholar 

  15. McKay, B.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)

    MathSciNet  MATH  Google Scholar 

  16. McKay, B., Piperno, A.: Practical graph isomorphism. II. Journal of Symbolic Computation 60, 94–112 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ostrowski, J.P., Linderoth, J., Rossi, F., Smriglio, S.: Constraint Orbital Branching. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 225–239. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  18. Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Math. Program. 126, 147–178 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. The GAP Group: GAP - Groups, Algorithms and Programming. Version 4.7.4 (2014)

    Google Scholar 

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Acknowledgments

The first author (GD) is financially supported by a CNPq Ph.D. thesis award.

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Authors and Affiliations

  1. CNRS LIX, Ecole Polytechnique, 91128, Palaiseau, France

    Gustavo Dias & Leo Liberti

Authors
  1. Gustavo Dias
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  2. Leo Liberti
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Corresponding author

Correspondence to Gustavo Dias .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

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Dias, G., Liberti, L. (2015). Orbital Independence in Symmetric Mathematical Programs. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_34

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  • DOI: https://doi.org/10.1007/978-3-319-26626-8_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26625-1

  • Online ISBN: 978-3-319-26626-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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