Compact representations of all members of an independence system

Article

Abstract

It is well known that (reduced, ordered) binary decision diagrams (BDDs) can sometimes be compact representations of the full solution set of Boolean optimization problems. Recently they have been suggested to be useful as discrete relaxations in integer and constraint programming (Hoda et al. 2010). We show that for every independence system there exists a top-down (i.e., single-pass) construction rule for the BDD. Furthermore, for packing and covering problems on n variables whose bandwidth is bounded by \(\mathcal {O}(\log n)\) the maximum width of the BDD is bounded by \(\mathcal {O}(n)\). We also characterize minimal widths of BDDs representing the set of all solutions to a stable set problem for various basic classes of graphs. Besides implicitly enumerating or counting all solutions and optimizing a class of nonlinear objective functions that includes separable functions, the results can be applied for effective evaluation of generating functions.

Keywords

Compact representations Boolean optimization Enumeration Stable sets Discrete relaxations Decision diagrams 

Mathematics Subject Classifications (2010)

52B99 90C27 05A15 05C30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amaldi, E., Bosio, S., Malucelli, F.: Hyperbolic set covering problems with competing ground-set elements. Math. Program. 134(2), 323–348 (2012). doi:10.1007/s10107-010-0431-1 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bergman, D., Cire, A.A., van Hoeve, W.J., Hooker, J.N.: Variable ordering for the application of BDDs to the maximum independent set problem. In: Proceedings of the 9th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR’12, pp. 34–49. Springer-Verlag, Berlin, Heidelberg (2012). doi:10.1007/978-3-642-29828-8_3 Google Scholar
  3. 3.
    Bergman, D., Cire, A.A., van Hoeve, W.J., Yunes, T.: BDD-based heuristics for binary optimization. J. Heuris. 20(2), 211–234 (2014). doi:10.1007/s10732-014-9238-1 CrossRefGoogle Scholar
  4. 4.
    Bergman, D., van Hoeve, W.J., Hooker, J.N.: Manipulating MDD relaxations for combinatorial optimization. In: Proceedings of the 8th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR’11, pp. 20–35. Springer-Verlag, Berlin, Heidelberg (2011)MATHGoogle Scholar
  5. 5.
    Bryant, R.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. C 35(8), 677–691 (1986). doi:10.1109/TC.1986.1676819 CrossRefMATHGoogle Scholar
  6. 6.
    Burkard, R., Ċela, E., Pardalos, P., Pitsoulis, L.: The quadratic assignment problem. In: Du, D.Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, vol. 3, pp. chap. 13, pp. 241–337. Kluwer Academic Publishers (1998)Google Scholar
  7. 7.
    Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 1969 24th National Conference, ACM ’69, pp. 157–172. ACM, New York (1969). doi:10.1145/800195.805928
  8. 8.
    Flajolet, P., Fusy, Ė., Pivoteau, C.: Boltzmann sampling of unlabeled structures. In: Panario, D., Sedgewick, R. (eds.) Proceedings of the Fourth Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2007, pp. 201–211. SIAM, New Orleans (2007). doi:10.1137/1.9781611972979.5
  9. 9.
    Hamacher, H., Queyranne, M.: K best solutions to combinatorial optimization problems. Ann. Oper. Res. 4(1), 123–143 (1985). doi:10.1007/BF02022039 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Harary, F.: Graph Theory. Addison-Wesley Series in Mathematics, Perseus Books. https://books.google.com/books?id=9nOljWrLzAAC (1994)
  11. 11.
    Haus, U.U., Köppe, M., Weismantel, R.: A primal all-integer algorithm based on irreducible solutions. Math. Program. Series B 96(2), 205–246 (2003). doi:10.1007/s10107-003-0384-8 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Henk, M., Köppe, M., Weismantel, R.: Integral decomposition of polyhedra and some applications in mixed integer programming. Math. Program. Series B 94(2–3), 193–206 (2003). doi:10.1007/s10107-002-0315-0
  13. 13.
    Hoda, S., van Hoeve, W.J., Hooker, J.: A systematic approach to MDD-based constraint programming. In: Cohen, D. (ed.) Principles and Practice of Constraint Programming – CP 2010, Lecture Notes in Computer Science, vol. 6308, pp. 266–280. Springer, Berlin Heidelberg (2010). doi:10.1007/978-3-642-15396-9_23 Google Scholar
  14. 14.
    Johnson, D.S., Trick, M.A. (eds.): Clique, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS, vol. 26. American Mathematical Society. http://dimacs.rutgers.edu/Volumes/Vol26.html (1996)
  15. 15.
    Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Syst. Tech. J. 38(4), 985–999 (1959)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design: OBDD – Foundations and Applications. Springer-Verlag New York, Inc., Secaucus (1998)CrossRefMATHGoogle Scholar
  17. 17.
    Resende, M.G.C.: Steiner triple covering problem instances. Available at http://www2.research.att.com/mgcr/data/index.html
  18. 18.
    SBCL community: Sbcl 1.1.5, an implementation of ANSI Common Lisp. Available as free software from http://www.sbcl.org
  19. 19.
    Schrijver, A.: Combinatorial Optimization. Springer, Berlin, Germany (2002)MATHGoogle Scholar
  20. 20.
    Sekine, K., Imai, H., Tani, S.: Computing the Tutte polynomial of a graph of moderate size. In: Staples, J., Eades, P., Katoh, N., Moffat, A. (eds.) Algorithms and Computations, Lecture Notes in Computer Science, vol. 1004, pp. 224–233. Springer, Berlin Heidelberg (1995). doi:10.1007/BFb0015427 Google Scholar
  21. 21.
    Yao, P.: Graph 2-isomorphism is NP-complete. Inf. Process. Lett. 9(2), 68–72 (1979). doi:10.1016/0020-0190(79)90130-3 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Cray EMEA Research LabBaselSwitzerland
  2. 2.WID, OptimizationUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations