Mathematical Programming

, Volume 132, Issue 1–2, pp 209–260 | Cite as

A polyhedral approach to the alldifferent system

Full Length Paper Series A

Abstract

This paper examines the facial structure of the convex hull of integer vectors satisfying a system of alldifferent predicates, also called an alldifferent system. The underlying analysis is based on a property, called inclusion, pertinent to such a system. For the alldifferent systems for which this property holds, we present two families of facet-defining inequalities, establish that they completely describe the convex hull and show that they can be separated in polynomial time. Consequently, the inclusion property characterises a group of alldifferent systems for which the linear optimization problem (i.e. the problem of optimizing a linear function over that system) can be solved in polynomial time. Furthermore, we establish that, for systems with three predicates, the inclusion property is also a necessary condition for the convex hull to be described by those two families of inequalities. For the alldifferent systems that do not possess that property, we establish another family of facet-defining inequalities and an accompanied polynomial-time separation algorithm. All the separation algorithms are incorporated within a cutting-plane scheme and computational experience on a set of randomly generated instances is reported. In concluding, we show that the pertinence of the inclusion property can be decided in polynomial time.

Keywords

alldifferent predicate Convex hull Separation Complexity 

Mathematics Subject Classification (2000)

90C10 90C27 90C35 90C57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Achterberg, T.: SCIP—a Framework to Integrate Constraint and Mixed Integer Programming. http://www.zib.de/Publications/Reports/ZR-04-19.pdf (2004)
  2. 2.
    Appa, G., Magos, D., Mourtos, I.: LP relaxations of multiple all_different predicates. In: Régin, J., Rueher, M. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization problems. 1st International Conference CPAIOR 2004, Nice, France, Lecturer Notes in Computer Science, vol. 3011, pp. 21–36 (2004)Google Scholar
  3. 3.
    Appa G., Magos D., Mourtos I.: On the system of two alldifferent predicates. Inf. Process. Lett. 94, 99–105 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Aron, I.D., Hooker, J.N., Yunes, T.H.: SIMPL: A system for integrating optimization techniques. In: Régin, J., Rueher, M., (eds.) Integration of AI and OR techniques in Constraint Programming for Combinatorial Optimization problems. 1st International Conference CPAIOR 2004, Nice, France, Lecturer Notes in Computer Science, vol. 3011, pp. 21–36 (2004)Google Scholar
  5. 5.
    Aron, I.D., Leventhal, D.H. Sellmann, M.: A Totally Unimodular Description of the Consistent Value Polytope for Binary Constraint Programming. In: Beck, J.C., Smith, B.M. (eds.) Integration of AI and OR techniques in Constraint Programming for Combinatorial Optimization problems. 3rd International Conference CPAIOR 2006, Cork, Ireland. Lecturer Notes in Computer Science, vol. 3990, pp. 16–28 (2006)Google Scholar
  6. 6.
    Balas E., Bockmayr A., Pisaruk N., Wolsey L.: On unions and dominants of polytopes. Math. program. 99, 223–239 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chaitin G.J., Auslander M., Chandra A.K., Cocke J., Hopkins M.E., Markstein P.: Register allocation via coloring. Comput. Lang. 6, 47–57 (1981)MATHCrossRefGoogle Scholar
  8. 8.
    Colombani, Y. Heipcke, S.: Mosel: An Overview. Dash Optimization (2002)Google Scholar
  9. 9.
    Elbassioni, K., Katriel, I., Kutz, M., Mahajan, M.: Simultaneous matchings. In: Deng, X., Du, D., (eds.) Algorithms and Computation. 16th International Symposium, ISAAC 2005, Sanya, Hainan, China, Lecturer Notes in Computer Science, vol. 3827, pp. 106–115 (2005)Google Scholar
  10. 10.
    Gomes, C.P., Shmoys, D.: The promise of LP to boost CSP techniques for combinatorial problems. In: Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization problems. 4th International Workshop CPAIOR 2002, Le Croisic, France, pp. 291–305 (2002)Google Scholar
  11. 11.
    Grötschel M., Lovász L., Schrijver A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hooker J.N.: Logic-Based Methods for Optimization. Wiley InterScience, New York (2000)MATHCrossRefGoogle Scholar
  13. 13.
    Hooker, J.N.: A search-infer-and-relax framework for integrating solution methods. In: Bartak, R., Milano, M., (eds.) Integration of AI and OR techniques in Constraint Programming for Combinatorial Optimization problems. 2nd International Conference CPAIOR 2005, Prague, Czech Republic, Lecturer Notes in Computer Science, vol. 3524, pp. 243–257 (2005)Google Scholar
  14. 14.
    Hooker J.N.: Logic-based modeling. In: Appa, G., Pitsoulis, L., Williams, H.P. (eds) Handbook on modelling for Discrete Optimization, International Series in Operations Research Management & Science, Springer, New York (2006)Google Scholar
  15. 15.
    Hooker J.N.: Integrated methods for optimization, International Series in Operations Research & Management Science. Springer, New York (2007)Google Scholar
  16. 16.
    Hooker J.N., Osorio M.A.: Mixed logical-linear programming. Discret. Appl. Math. 96(97), 395–442 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hooker, J.N., Yan, H.: A relaxation of the cumulative constraint. In: van Hentenryck, P. (ed.) Principles and Practice of Constraint Programming, 8th International Conference, CP2002, Ithaka, NY, Lecturer Notes in Computer Science, vol. 2470, pp. 80–92 (2002)Google Scholar
  18. 18.
    ILOG S.A., ILOG CPLEX Callable Library 9.1 (2005)Google Scholar
  19. 19.
    Kaya, L.G., Hooker, J.N.: The circuit polytope. http://wpweb2.tepper.cmu.edu/jnh/CircuitPolytope.pdf
  20. 20.
    Kaznachey D., Jagota A., Das S.: Neural network-based heuristic algorithms for hypergraph coloring problems with applications. J. Parallel Distrib. Comput. 63, 786–800 (2003)MATHCrossRefGoogle Scholar
  21. 21.
    Lee J.: All-different polytopes. J. Comb. Opt. 6, 335–352 (2002)MATHCrossRefGoogle Scholar
  22. 22.
    Milano M., Ottosson G., Refalo P., Thorsteinsson E.: The role of integer programming techniques in constraint programming’s global constraints. Inf. J. Comput. 14(4), 387–402 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mirsky, L.: Transversal theory, Mathematics Science and Engineering. vol. 75, Academic Press, London (1971)Google Scholar
  24. 24.
    Refalo, P.: Linear formulation of constraint programming models and hybrid solvers. In: Dechter, R. (ed.) Principles and Practice of Constraint Programming, 6th International Conference, CP 2000, Singapore, Lecturer Notes in Computer Science, vol. 1894, pp. 369–383 (2000)Google Scholar
  25. 25.
    Régin J.C.: Cost-based arc consistency for global cardinality constraints. Constraints 7, 387–405 (2002)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Régin, J.C., Gomes, C.P.: The cardinality matrix constraint. In: Wallace, M. (ed.) Principles and Practice of Constraint Programming, 10th International Conference, CP 2006, Toronto, Lecturer Notes in Computer Science, vol. 3258, pp. 572–587 (2004)Google Scholar
  27. 27.
    Sellmann, M., Mercier, L., Leventhal, D.H.: The linear programming polytope of binary constraint problems with bounded tree-width. In: Van Hentenryck, P., Wolsey, L. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization problems. 4th International Conference CPAIOR 2007, Brussels, Belgium. Lecturer Notes in Computer Science, vol. 4510, pp. 275–287 (2007)Google Scholar
  28. 28.
    van Hentenryck P.: The OPL Optimization Programming Language. MIT Press, Boston (1999)Google Scholar
  29. 29.
    van Hoeve, W.J.: The Alldifferent constraint: A survey. Sixth Annual Workshop of the ERCIM Working Group on Constraints, Prague (2001)Google Scholar
  30. 30.
    Williams H.P., Yan H.: Representations of the all-different predicate of constraint satisfaction in integer programming. Inf. J. Comput. 13, 96–103 (2001)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yan H., Hooker J.N.: Tight representation of logic constraints as cardinality rules. Math. Program. 85, 363–377 (1999)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Yunes, T.H.: On the sum constraint: relaxation and applications. In: van Hentenryck, P. (ed.) Principles and Practice of Constraint Programming, 8th International Conference, CP2002, Ithaka, NY, Lecturer Notes in Computer Science, vol. 2470, pp. 80–92 (2002)Google Scholar

Copyright information

© Springer and Mathematical Optimization Society 2010

Authors and Affiliations

  1. 1.Department of InformaticsTechnological Educational Institute of AthensEgaleoGreece
  2. 2.Department of Management Science and TechnologyAthens University of Economics and BusinessAthensGreece
  3. 3.Department of Management, Operational Research Group (ORG)London School of EconomicsLondonUK

Personalised recommendations