Advertisement

Quadratic Reformulation of Nonlinear Pseudo-Boolean Functions via the Constraint Composite Graph

  • Ka Wa YipEmail author
  • Hong XuEmail author
  • Sven Koenig
  • T. K. Satish Kumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11494)

Abstract

Nonlinear pseudo-Boolean optimization (nonlinear PBO) is the minimization problem on nonlinear pseudo-Boolean functions (nonlinear PBFs). One promising approach to nonlinear PBO is to first use a quadratization algorithm to reduce the PBF to a quadratic PBF by introducing intelligently chosen auxiliary variables and then solve it using a quadratic PBO solver. In this paper, we develop a new quadratization algorithm based on the idea of the constraint composite graph (CCG). We demonstrate its theoretical advantages over state-of-the-art quadratization algorithms. We experimentally demonstrate that our CCG-based quadratization algorithm outperforms the state-of-the-art algorithms in terms of both effectiveness and efficiency on randomly generated instances and a novel reformulation of the uncapacitated facility location problem.

References

  1. 1.
    Abío, I., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E., Mayer-Eichberger, V.: A new look at BDDs for Pseudo-Boolean constraints. J. Artif. Intell. Res. 45(1), 443–480 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    AlBdaiwi, B.F., Goldengorin, B., Sierksma, G.: Equivalent instances of the simple plant location problem. Comput. Math. Appl. 57(5), 812–820 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anthony, M., Boros, E., Crama, Y., Gruber, A.: Quadratization of symmetric Pseudo-Boolean functions. Discrete Appl. Math. 203, 1–12 (2016).  https://doi.org/10.1016/j.dam.2016.01.001MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anthony, M., Boros, E., Crama, Y., Gruber, A.: Quadratic reformulations of nonlinear binary optimization problems. Math. Program. 162(1–2), 115–144 (2017).  https://doi.org/10.1007/s10107-016-1032-4MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beresnev, V.: On a problem of mathematical standardization theory. Upr. Sistemy 11, 43–54 (1973). (in Russian)Google Scholar
  6. 6.
    Berthold, T., Heinz, S., Pfetsch, M.E.: Nonlinear Pseudo-Boolean optimization: relaxation or propagation? In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 441–446. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02777-2_40CrossRefGoogle Scholar
  7. 7.
    Bockmayr, A.: Logic programming with Pseudo-Boolean constraints. In: Constraint Logic Programming, pp. 327–350 (1993)Google Scholar
  8. 8.
    Bofill, M., Coll, J., Suy, J., Villaret, M.: Compact MDDs for Pseudo-Boolean constraints with at-most-one relations in resource-constrained scheduling problems. In: International Joint Conference on Artificial Intelligence, pp. 555–562 (2017).  https://doi.org/10.24963/ijcai.2017/78
  9. 9.
    Boros, E., Crama, Y., Rodríguez-Heck, E.: Quadratizations of symmetric Pseudo-Boolean functions: Sub-linear bounds on the number of auxiliary variables. In: International Symposium on Artificial Intelligence and Mathematics (2018). http://isaim2018.cs.virginia.edu/papers/ISAIM2018_Boolean_Boros_etal.pdf
  10. 10.
    Boros, E., Gruber, A.: On quadratization of Pseudo-Boolean functions (2014). arXiv preprint: arXiv:1404.6538
  11. 11.
    Boros, E., Hammer, P.L., Minoux, M., Rader Jr., D.J.: Optimal cell flipping to minimize channel density in VLSI design and Pseudo-Boolean optimization. Discrete Appl. Math. 90(1–3), 69–88 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7(5), 193–209 (2008).  https://doi.org/10.1007/s11128-008-0082-9MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eén, N., Sörensson, N.: Translating Pseudo-Boolean constraints into SAT. J. Satisf. Boolean Model. Comput. 2, 1–26 (2006)zbMATHGoogle Scholar
  14. 14.
    Fioretto, F., Xu, H., Koenig, S., Kumar, T.K.S.: Solving multiagent constraint optimization problems on the constraint composite graph. In: Miller, T., et al. (eds.) PRIMA 2018. LNCS (LNAI), vol. 11224, pp. 106–122. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-03098-8_7CrossRefGoogle Scholar
  15. 15.
    Freeman, R.J., Gogerty, D.C., Graves, G.W., Brooks, R.B.: A mathematical model of supply support for space operations. Oper. Res. 14(1), 1–15 (1966)CrossRefGoogle Scholar
  16. 16.
    Glover, F., Woolsey, E.: Converting the 0-1 polynomial programming problem to a 0-1 linear program. Oper. Res. 22(1), 180–182 (1974)CrossRefGoogle Scholar
  17. 17.
    Gruber, A.G.: Algorithmic and complexity results for Boolean and Pseudo-Boolean functions. Ph.D. thesis, Rutgers University-Graduate School-New Brunswick (2015)Google Scholar
  18. 18.
    Hammer, P.L., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas, vol. 7. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  19. 19.
    Hammer, P.: Plant location – a Pseudo-Boolean approach. Isr. J. Technol. 6, 330–332 (1968)zbMATHGoogle Scholar
  20. 20.
    Hansen, P., Kochetov, Y., Mladenovi, N.: Lower bounds for the uncapacitated facility location problem with user preferences. Groupe d’études et de recherche en analyse des décisions, HEC Montréal (2004)Google Scholar
  21. 21.
    Ishikawa, H.: Transformation of general binary MRF minimization to the first-order case. IEEE Trans. Pattern Anal. Mach. Intell. 33(6), 1234–1249 (2011).  https://doi.org/10.1109/TPAMI.2010.91CrossRefGoogle Scholar
  22. 22.
    Johnson, M.W., et al.: Quantum annealing with manufactured spins. Nature 473, 194–198 (2011).  https://doi.org/10.1038/nature10012CrossRefGoogle Scholar
  23. 23.
    Joshi, S., Martins, R., Manquinho, V.: Generalized totalizer encoding for Pseudo-Boolean constraints. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 200–209. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-23219-5_15CrossRefGoogle Scholar
  24. 24.
    Kahl, F., Strandmark, P.: Generalized roof duality for Pseudo-Boolean optimization. In: International Conference on Computer Vision, pp. 255–262 (2011).  https://doi.org/10.1109/ICCV.2011.6126250
  25. 25.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts – a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1274–1279 (2007).  https://doi.org/10.1109/TPAMI.2007.1031CrossRefGoogle Scholar
  27. 27.
    Kumar, T.K.S.: Incremental computation of resource-envelopes in producer-consumer models. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 664–678. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45193-8_45CrossRefGoogle Scholar
  28. 28.
    Kumar, T.K.S.: A framework for hybrid tractability results in boolean weighted constraint satisfaction problems. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 282–297. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85958-1_19CrossRefGoogle Scholar
  29. 29.
    Kumar, T.K.S.: Lifting techniques for weighted constraint satisfaction problems. In: International Symposium on Artificial Intelligence and Mathematics (2008). http://isaim2008.unl.edu/PAPERS/TechnicalProgram/ISAIM2008_0004_d1de5114b3cb94de7e670ab2905c3b3d.pdf
  30. 30.
    Kumar, T.K.S.: Kernelization, generation of bounds, and the scope of incremental computation for weighted constraint satisfaction problems. In: International Symposium on Artificial Intelligence and Mathematics (2016)Google Scholar
  31. 31.
    Manquinho, V., Marques-Silva, J., Planes, J.: Algorithms for weighted Boolean optimization. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 495–508. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02777-2_45CrossRefGoogle Scholar
  32. 32.
    Manquinho, V., Roussel, O.: Pseudo-Boolean competition (2016). http://www.cril.univ-artois.fr/PB16
  33. 33.
    Manquinho, V.M., Marques-Silva, J.: On using cutting planes in Pseudo-Boolean optimization. J. Satisf. Boolean Model. Comput. 2, 209–219 (2006)zbMATHGoogle Scholar
  34. 34.
    Osokin, A.: MATLAB wrapper to the QPBO algorithm by V. Kolmogorov (2014). https://github.com/aosokin/qpboMex
  35. 35.
    Philipp, T., Steinke, P.: PBLib – a library for encoding Pseudo-Boolean constraints into CNF. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 9–16. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24318-4_2CrossRefGoogle Scholar
  36. 36.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121(2), 307 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rhys, J.: A selection problem of shared fixed costs and network flows. Manag. Sci. 17(3), 200–207 (1970)CrossRefGoogle Scholar
  38. 38.
    Wegener, I., Witt, C.: On the analysis of a simple evolutionary algorithm on quadratic Pseudo-Boolean functions. J. Discrete Algorithms 3(1), 61–78 (2005)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Xu, H., Koenig, S., Kumar, T.K.S.: A constraint composite graph-based ILP encoding of the Boolean weighted CSP. In: Beck, J.C. (ed.) CP 2017. LNCS, vol. 10416, pp. 630–638. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66158-2_40CrossRefGoogle Scholar
  40. 40.
    Xu, H., Satish Kumar, T.K., Koenig, S.: The Nemhauser-Trotter reduction and lifted message passing for the weighted CSP. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 387–402. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-59776-8_31CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

Personalised recommendations