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Solving Multiobjective Discrete Optimization Problems with Propositional Minimal Model Generation

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Principles and Practice of Constraint Programming (CP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10416))

Abstract

We propose a propositional logic based approach to solve MultiObjective Discrete Optimization Problems (MODOPs). In our approach, there exists a one-to-one correspondence between a Pareto front point of MODOP and a \(P\)-minimal model of the CNF formula obtained from MODOP. This correspondence is achieved by adopting the order encoding as CNF encoding for multiobjective functions. Finding the Pareto front is done by enumerating all P-minimal models. The beauty of the approach is that each Pareto front point is blocked by a single clause that contains at most one literal for each objective function. We evaluate the effectiveness of our approach by empirically contrasting it to a state-of-the-art MODOP solving technique.

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Notes

  1. 1.

    http://bach.istc.kobe-u.ac.jp/sugar/.

  2. 2.

    MODOP other than multiobjective functions can be encoded by existing CNF encodings: direct, multivalue, support, log, and hybrid encodings.

  3. 3.

    Note that there is no essential differences between them.

  4. 4.

    The terminology “block” is often used in SAT/MaxSAT communities to denote adding a constraint which prevents a solution to be found again (to “block” it) in an iterative process.

  5. 5.

    https://potassco.org/.

  6. 6.

    #Sets = 20 is used only for #Items = 100. #Sets = 30 is used only for #Items = 150.

  7. 7.

    http://www.andrew.cmu.edu/user/vanhoeve/mdd/.

  8. 8.

    We confirmed in our computational environment that the BDD solver was able to solve 55 instances out of the 60 MSCP instances, compared with 53 in [6].

  9. 9.

    http://kix.istc.kobe-u.ac.jp/~soh/sucre/.

References

  1. Alarcon-Rodriguez, A., Ault, G., Galloway, S.: Multi-objective planning of distributed energy resources: a review of the state-of-the-art. Renew. Sustain. Energy Rev. 14(5), 1353–1366 (2010)

    Article  Google Scholar 

  2. Ansótegui, C., Manyà, F.: Mapping problems with finite-domain variables to problems with boolean variables. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 1–15. Springer, Heidelberg (2005). doi:10.1007/11527695_1

    Chapter  Google Scholar 

  3. Bailleux, O., Boufkhad, Y.: Efficient CNF encoding of boolean cardinality constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 108–122. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45193-8_8

    Chapter  Google Scholar 

  4. Ballestero, E., Bravo, M., Pérez-Gladish, B., Parra, M.A., Plà-Santamaria, D.: Socially responsible investment: a multicriteria approach to portfolio selection combining ethical and financial objectives. Eur. J. Oper. Res. 216(2), 487–494 (2012)

    Article  MathSciNet  Google Scholar 

  5. Banbara, M., Matsunaka, H., Tamura, N., Inoue, K.: Generating combinatorial test cases by efficient SAT encodings suitable for CDCL SAT solvers. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 112–126. Springer, Heidelberg (2010). doi:10.1007/978-3-642-16242-8_9

    Chapter  Google Scholar 

  6. Bergman, D., Cire, A.A.: Multiobjective optimization by decision diagrams. In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 86–95. Springer, Cham (2016). doi:10.1007/978-3-319-44953-1_6

    Chapter  Google Scholar 

  7. Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications (FAIA), vol. 185. IOS Press, Amsterdam (2009)

    Google Scholar 

  8. Boland, N., Charkhgard, H., Savelsbergh, M.W.P.: A new method for optimizing a linear function over the efficient set of a multiobjective integer program. Eur. J. Oper. Res. 260(3), 904–919 (2017)

    Article  MathSciNet  Google Scholar 

  9. Brewka, G., Delgrande, J.P., Romero, J., Schaub, T.: asprin: customizing answer set preferences without a headache. In: Proceedings of the 29th National Conference on Artificial Intelligence (AAAI 2015), pp. 1467–1474 (2015)

    Google Scholar 

  10. Burke, E.K., Li, J., Qu, R.: A pareto-based search methodology for multi-objective nurse scheduling. Ann. Oper. Res. 196(1), 91–109 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crawford, J.M., Baker, A.B.: Experimental results on the application of satisfiability algorithms to scheduling problems. In: Proceedings of the 12th National Conference on Artificial Intelligence (AAAI 1994), pp. 1092–1097 (1994)

    Google Scholar 

  12. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Schoenauer, M., Deb, K., Rudolph, G., Yao, X., Lutton, E., Merelo, J.J., Schwefel, H.-P. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 849–858. Springer, Heidelberg (2000). doi:10.1007/3-540-45356-3_83

    Chapter  Google Scholar 

  13. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  14. Ehrgott, M.: Multicriteria Optimization. Springer, Heidelberg (2005). doi:10.1007/3-540-27659-9

    MATH  Google Scholar 

  15. Figueira, J., Greco, S., Ehrgott, M.: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science. Springer, Heidelberg (2005). doi:10.1007/b100605

    Book  MATH  Google Scholar 

  16. Gent, I.P., Nightingale, P.: A new encoding of alldifferent into SAT. In: Proceedings of the 3rd International Workshop on Modelling and Reformulating Constraint Satisfaction Problems (2004)

    Google Scholar 

  17. Inoue, K., Soh, T., Ueda, S., Sasaura, Y., Banbara, M., Tamura, N.: A competitive and cooperative approach to propositional satisfiability. Discrete Appl. Math. 154(16), 2291–2306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iturriaga, S., Dorronsoro, B., Nesmachnow, S.: Multiobjective evolutionary algorithms for energy and service level scheduling in a federation of distributed datacenters. Int. Trans. Oper. Res. 24(1–2), 199–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kirlik, G., Sayin, S.: A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems. Eur. J. Oper. Res. 232(3), 479–488 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Knuth, D.E.: The Art of Computer Programming, Volume 4, Fascicle 6: Satisfiability. Addison-Wesley Professional, Boston (2015)

    Google Scholar 

  21. Koshimura, M., Nabeshima, H., Fujita, H., Hasegawa, R.: Minimal model generation with respect to an atom set. In: Proceedings of the the 7th International Workshop on First-Order Theorem Proving (FTP 2009), pp. 49–59 (2009)

    Google Scholar 

  22. Laumanns, M., Thiele, L., Zitzler, E.: An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur. J. Oper. Res. 169(3), 932–942 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lukasiewycz, M., Glaß, M., Haubelt, C., Teich, J.: Solving multi-objective pseudo-boolean problems. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 56–69. Springer, Heidelberg (2007). doi:10.1007/978-3-540-72788-0_9

    Chapter  Google Scholar 

  24. Marinescu, R.: Exploiting problem decomposition in multi-objective constraint optimization. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 592–607. Springer, Heidelberg (2009). doi:10.1007/978-3-642-04244-7_47

    Chapter  Google Scholar 

  25. Marinescu, R.: Best-first vs. depth-first AND/OR search for multi-objective constraint optimization. In: Proceedings of the 22nd IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2010), pp. 439–446 (2010)

    Google Scholar 

  26. Metodi, A., Codish, M., Lagoon, V., Stuckey, P.J.: Boolean equi-propagation for optimized SAT encoding. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 621–636. Springer, Heidelberg (2011). doi:10.1007/978-3-642-23786-7_47

    Chapter  Google Scholar 

  27. Nabeshima, H., Soh, T., Inoue, K., Iwanuma, K.: Lemma reusing for SAT based planning and scheduling. In: Proceedings of the International Conference on Automated Planning and Scheduling 2006 (ICAPS 2006), pp. 103–112 (2006)

    Google Scholar 

  28. Niemelä, I.: A tableau calculus for minimal model reasoning. In: Miglioli, P., Moscato, U., Mundici, D., Ornaghi, M. (eds.) TABLEAUX 1996. LNCS, vol. 1071, pp. 278–294. Springer, Heidelberg (1996). doi:10.1007/3-540-61208-4_18

    Chapter  Google Scholar 

  29. Ohrimenko, O., Stuckey, P.J., Codish, M.: Propagation via lazy clause generation. Constraints 14(3), 357–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Okimoto, T., Joe, Y., Iwasaki, A., Matsui, T., Hirayama, K., Yokoo, M.: Interactive algorithm for multi-objective constraint optimization. In: Milano, M. (ed.) CP 2012. LNCS, pp. 561–576. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33558-7_41

    Chapter  Google Scholar 

  31. Ozlen, M., Burton, B.A., MacRae, C.A.G.: Multi-objective integer programming: an improved recursive algorithm. J. Optim. Theory Appl. 160(2), 470–482 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), pp. 86–92 (2000)

    Google Scholar 

  33. Rollon, E., Larrosa, J.: Bucket elimination for multiobjective optimization problems. J. Heuristics 12(4–5), 307–328 (2006)

    Article  MATH  Google Scholar 

  34. Rollon, E., Larrosa, J.: Multi-objective russian doll search. In: Proceedings of the 22nd National Conference on Artificial Intelligence (AAAI 2007), pp. 249–254 (2007)

    Google Scholar 

  35. Schwind, N., Okimoto, T., Konieczny, S., Wack, M., Inoue, K.: Utilitarian and egalitarian solutions for multi-objective constraint optimization. In: Proceedings of the 26th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2014), pp. 170–177. IEEE Computer Society (2014)

    Google Scholar 

  36. Soh, T., Banbara, M., Tamura, N.: Proposal and evaluation of hybrid encoding of CSP to SAT integrating order and log encodings. Int. J. Artif. Intell. Tools 26(1), 1–29 (2017)

    Article  Google Scholar 

  37. Soh, T., Inoue, K., Tamura, N., Banbara, M., Nabeshima, H.: A SAT-based method for solving the two-dimensional strip packing problem. Fundam. Inf. 102(3–4), 467–487 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Tamura, N., Banbara, M., Soh, T.: PBSugar: Compiling pseudo-boolean constraints to SAT with order encoding. In: Proceedings of the 25th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2013), pp. 1020–1027. IEEE, November 2013

    Google Scholar 

  39. Tamura, N., Taga, A., Kitagawa, S., Banbara, M.: Compiling finite linear CSP into SAT. Constraints 14(2), 254–272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tanjo, T., Tamura, N., Banbara, M.: Sugar++: a SAT-based Max-CSP/COP solver. In: Proceedings of the 3rd International CSP Solver Competition, pp. 77–82 (2008)

    Google Scholar 

  41. Ugarte, W., Boizumault, P., Crémilleux, B., Lepailleur, A., Loudni, S., Plantevit, M., Raïssi, C., Soulet, A.: Skypattern mining: from pattern condensed representations to dynamic constraint satisfaction problems. Artif. Intell. 244, 48–69 (2017). https://doi.org/10.1016/j.artint.2015.04.003

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, L., Ng, A.H.C., Deb, K. (eds.): Multi-objective Evolutionary Optimisation for Product Design and Manufacturing. Springer, Heidelberg (2011). doi:10.1007/978-0-85729-652-8

    Google Scholar 

  43. Wilson, N., Razak, A., Marinescu, R.: Computing possibly optimal solutions for multi-objective constraint optimisation with tradeoffs. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), pp. 815–822 (2015)

    Google Scholar 

  44. Yi, D., Goodrich, M.A., Seppi, K.D.: MORRF*: sampling-based multi-objective motion planning. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), pp. 1733–1741 (2015)

    Google Scholar 

  45. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: improving the strength pareto evolutionary algorithm for multiobjective optimization. In: Proceedings of the 4th International Conference of Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001), pp. 95–100 (2002)

    Google Scholar 

  46. Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Proceedings of the 8th International Conference on Parallel Problem Solving from Nature (PPSN VIII), pp. 832–842 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Information Science and Technology Center, Kobe University, Kobe, Japan

    Takehide Soh, Mutsunori Banbara & Naoyuki Tamura

  2. CRIL-CNRS, Université d’Artois, Lens, France

    Daniel Le Berre

Authors
  1. Takehide Soh
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  2. Mutsunori Banbara
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  3. Naoyuki Tamura
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  4. Daniel Le Berre
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Correspondence to Takehide Soh .

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Editors and Affiliations

  1. University of Toronto, Toronto, Ontario, Canada

    J. Christopher Beck

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Soh, T., Banbara, M., Tamura, N., Le Berre, D. (2017). Solving Multiobjective Discrete Optimization Problems with Propositional Minimal Model Generation. In: Beck, J. (eds) Principles and Practice of Constraint Programming. CP 2017. Lecture Notes in Computer Science(), vol 10416. Springer, Cham. https://doi.org/10.1007/978-3-319-66158-2_38

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  • DOI: https://doi.org/10.1007/978-3-319-66158-2_38

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