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An effective variable selection heuristic in SLS for weighted Max-2-SAT

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Abstract

Stochastic local search (SLS) is an appealing method for solving the maximum satisfiability (Max-SAT) problem. This paper proposes a new variable selection heuristic for Max-SAT local search algorithms, which works particularly well for weighted Max-2-SAT instances. Evolving from the recent configuration checking strategy, this new heuristic works in three levels and is called CCTriplex. According to the CCTriplex heuristic, a variable that is both decreasing and configuration changed has the higher priority to be flipped than a decreasing variable, which in turn has the higher priority than a configuration changed variable. The CCTriplex heuristic is used to develop a new SLS algorithm for weighted Max-2-SAT called CCMaxSAT. We evaluate CCMaxSAT on random benchmarks with different densities, and the hand crafted Frb benchmark, as well as weighted Max-2-SAT instances encoded from MaxCut, MaxClique and sports scheduling problems. Compared with the state-of-the-art SLS solver for weighted Max-2-SAT called ITS and the best SLS solver in Max-SAT Evaluation 2012 namely ubcsat-IRoTS, as well as the famous complete solver wMaxSATz, our algorithm CCMaxSAT shows rather good performance on all the benchmarks.

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Notes

  1. http://www.proin.ktu.lt/~gintaras/wmax2sat.html.

  2. http://ubcsat.dtompkins.com/downloads.

  3. http://home.mis.u-picardie.fr/~cli/EnglishPage.html.

  4. ftp://dimacs.rutgers.edu/pub/challenge/satisfiability/cotributed/selman/.

  5. http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/max-sat-benchmarks.htm.

  6. http://maxsat.ia.udl.cat:81/12/benchmarks/index.html.

  7. ftp://dimacs.rutgers.edu/pub/challenges.

  8. http://www-or.amp.i.kyoto-u.ac.jp/~yagiura/sat/max2sat/.

  9. www.shaoweicai.net/MaxSAT.html.

  10. http://www-or.amp.i.kyoto-u.ac.jp/~yagiura/sat/max2sat/.

References

  • Ansótegui, C., Bonet, M.L., Levy, J.: SAT-based MaxSAT algorithms. Artif. Intell. 196, 77–105 (2013)

    Article  MATH  Google Scholar 

  • Cai, S., Su, K.: Local search with configuration checking for SAT. In: Proceeding of the ICTAI-11, pp. 59–66 (2011)

  • Cai, S., Su, K.: Configuration checking with aspiration in local search for SAT. In: Proceedings of the AAAI-12, pp. 434–440 (2012)

  • Cai, S., Su, K.: Local search for Boolean satisfiability with configuration checking and subscore. Artif. Intell. 204, 75–98 (2013)

    Article  Google Scholar 

  • Cai, S., Su, K., Sattar, A.: Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artif. Intell. 175(9–10), 1672–1696 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Cai, S., Su, K., Luo, C., Sattar, A.: NuMVC: an efficient local search algorithm for minimum vertex cover. J. Artif. Intell. Res. 46, 687–716 (2013)

    MATH  MathSciNet  Google Scholar 

  • Dimitropoulos, X., Krioukov, D., Fomenkov, M., Huffaker, B., Hyun, Y., Claffy, K., Riley, G.: As relationships: inference and validation. Comput. Commun. Rev. 37(1), 29–40 (2007)

    Article  Google Scholar 

  • Festa, P., Pardalos, P., Pitsoulis, L., Resende, M.: GRASP with path relinking for the weighted Max-SAT problem. ACM J. Exp. Algorithmics (11) (2006)

  • Glover, F.: Tabu search—part I. INFORMS J. Comput. 1(3), 190–206 (1989)

    Article  MATH  Google Scholar 

  • Gramm, J., Hirsch, E., Niedermeier, R., Rossmanith, P.: Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT. Discret. Appl. Math. 130(2), 139–155 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Grosso, A., Locatelli, M., Pullan, W.: Simple ingredients leading to very efficient heuristics for the maximum clique problem. J. Heuristics 14(6), 587–612 (2008)

    Article  Google Scholar 

  • Haanpää, H., Kaski, P.: The near resolvable 2-(13, 4, 3) designs and thirteen-player whist tournaments. Des. Codes. Cryptogr. 35(3), 271–285 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Heras, F., Bañeres, D.: The impact of Max-SAT resolution-based preprocessors on local search solvers. J. Satisf. Boolean Model. Comput. 7, 89–126 (2010)

    MathSciNet  Google Scholar 

  • Heras, F., Larrosa, J., Oliveras, A.: MiniMax-SAT: an efficient weighted max-sat solver. J. Artif. Intell. Res. (JAIR) 31, 1–32 (2008)

    MATH  MathSciNet  Google Scholar 

  • Hoos, H.H.: On the Run-time Behaviour of Stochastic Local Search Algorithms for SAT. In: Proceedings of the AAAI-99, pp. 661–666 (1999)

  • Hoos, H.H.: An adaptive noise mechanism for WalkSAT. In: Proceedings of the AAAI-02, pp. 655–660 (2002)

  • Hoos, H.H., Stützle, T.: Stochastic Local Search: Foundations & Applications. Elsevier, Amsterdam (2004)

    Google Scholar 

  • Hutter, F., Tompkins, D.A.D., Hoos, H.H.: Scaling and probabilistic smoothing: efficient dynamic local search for SAT. In: Proceedings of the CP-02, pp. 233–248 (2002)

  • Janhunen, T., Niemelä, I., Seipel, D., Simons, P., You, J.H.: Unfolding partiality and disjunctions in stable model semantics. ACM Trans. Comput. Log. 7(1), 1–37 (2006)

    Article  MathSciNet  Google Scholar 

  • Kastner, R., Bozorgzadeh, E., Sarrafzadeh, M.: Pattern routing: use and theory for increasing predictability and avoiding coupling. In: IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, pp. 233–248 (2002)

  • Kochenberger, G., Glover, F., Alidaee, B., Lewis, K.: Using the unconstrained quadratic program to model and solve Max-2-SAT problems. Int. J. Oper. Res. 1, 89–100 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Kroc, L., Sabharwal, A., Gomes, C., Selman, B.: Integrating systematic and local search paradigms: a new strategy for MaxSAT. In: Proceedings of the IJCAI-09, pp. 544–551 (2009)

  • Li, C., Huang, W.: Diversification and determinism in local search for satisfiability. In: Proceedings of the SAT-05, pp. 158–172 (2005)

  • Li, C., Manyà, F., Planes, J.: New inference rules for Max-SAT. J. Artif. Intell. Res. (JAIR) 30, 321–359 (2007)

    Google Scholar 

  • Li, C., Manyà, F., Mohamedou, N., Planes, J.: Exploiting cycle structures in Max-SAT. In: Proceedings of the SAT-09, pp. 467–480 (2009)

  • Li, C., Wei, W., Zhang, H.: Combining adaptive noise and look-ahead in local search for SAT. In: Proceedings of the SAT-07, pp. 121–133 (2007)

  • Lin, H., Su, K., Li, C.: Within-problem learning for efficient lower bound computation in max-sat solving. In: Proceedings of the AAAI-08, pp. 351–356 (2008)

  • Littman, M.L., Majercik, S.M., Pitassi, T.: Stochastic boolean satisfiability. J. Autom. Reason. 27(3), 251–296 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Luo, C., Su, K., Cai, S.: Improving local search for random 3-SAT using quantitative configuration checking. In: Proceedings of the ECAI-12, pp. 570–575 (2012)

  • Luo, C., Cai, S., Wu, W., Su, K.: Focused random walk with configuration checking and break minimum for satisfiability. In: Proceedings of the CP-13, pp. 481–496 (2013)

  • Palubeckis, G.: Solving the weighted Max-2-SAT problem with iterated tabu search. J. Inf. Technol. Control 37, 275–284 (2008)

    Google Scholar 

  • Pullan, W., Mascia, F., Brunato, M.: Cooperating local search for the maximum clique problem. J. Heuristics 17(2), 181–199 (2011)

    Article  Google Scholar 

  • Ryuhei, M., Tomomi, M.: Semidefinite programming based approaches to the break minimization problem. Comput. OR 33, 1975–1982 (2006)

    Article  MATH  Google Scholar 

  • Selman, B., Kautz, H., Cohen, B.: Noise strategies for improving local search. In: Proceedings of the AAAI-94, pp. 337–343 (1994)

  • Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artif. Intell. 138(1–2), 181–234 (2002)

    Article  MATH  Google Scholar 

  • Smyth, K., Hoos, H.H., Stützle, T.: Iterated robust tabu search for MAX-SAT. In: Proceedings of the Canadian Conference on AI, pp. 129–144 (2003)

  • Staub, R., Prautzsch, H.: Creating optimized cutout sheets for paper models from meshes. In: Ninth SIAM Conference on Geometric Design and Computing (2005)

  • Wu, Z., Wah, B.W.: An efficient global-search strategy in discrete lagrangian methods for solving hard satisfiability problems. In: Proceedings of the AAAI-00, pp. 310–315 (2000)

  • Xu, K., Boussemart, F., Hemery, F., Lecoutre, C.: A simple model to generate hard satisfiable instances. In: Proceedings of the IJCAI-05, pp. 337–342 (2005)

  • Xu, K., Boussemart, F., Hemery, F., Lecoutre, C.: Random constraint satisfaction: easy generation of hard (satisfiable) instances. Artif. Intell. 171(8–9), 514–534 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments, which helped to improve the quality of this article. This work is supported by China National 973 project 2014CB340301, ARC Future Fellowship FT0991785, and National Natural Science Foundation of China 61370072 and 61472369.

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Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Shaowei Cai

  2. Key Laboratory of High Confidence Software Technologies, Peking University, Beijing, China

    Zhong Jie

  3. Institute for Integrated and Intelligent Systems, Griffith University, Brisbane, Australia

    Kaile Su

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  1. Shaowei Cai
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  2. Zhong Jie
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  3. Kaile Su
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Correspondence to Shaowei Cai.

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Cai, S., Jie, Z. & Su, K. An effective variable selection heuristic in SLS for weighted Max-2-SAT. J Heuristics 21, 433–456 (2015). https://doi.org/10.1007/s10732-015-9284-3

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