Skip to main content
SpringerLink
Log in

Modified extremal optimization for the hard maximum satisfiability problem

  • Published:
Journal of Zhejiang University SCIENCE C Aims and scope Submit manuscript

Abstract

Based on our recent study on probability distributions for evolution in extremal optimization (EO), we propose a modified framework called EOSAT to approximate ground states of the hard maximum satisfiability (MAXSAT) problem, a generalized version of the satisfiability (SAT) problem. The basic idea behind EOSAT is to generalize the evolutionary probability distribution in the Bose-Einstein-EO (BE-EO) algorithm, competing with other popular algorithms such as simulated annealing and WALKSAT. Experimental results on the hard MAXSAT instances from SATLIB show that the modified algorithms are superior to the original BE-EO algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alava, M., Ardelius, J., Aurell, E., Kaski, P., Krishnamurthy, S., Orponen, P., Seitz, S., 2008. Circumspect descent prevails in solving random constraint satisfaction problems. PNAS, 105(40):15253–15257. [doi:10.1073/pnas.0712263105]

    Article  Google Scholar 

  • Altarelli, F., Monasson, R., Smerjian, G., Zamponi, F., 2009. A Review of the Statistical Mechanics Approach to Random Optimization Problems. Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, Vol. 185. IOS Press, the Netherlands.

    Google Scholar 

  • Bak, P., Sneppen, K., 1993. Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett., 71(24):4083–4086. [doi:10.1103/PhysRevLett.71.4083]

    Article  Google Scholar 

  • Bak, P., Tang, C., Wiesenfeld, K., 1987. Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett., 59(4):381–384. [doi:10.1103/PhysRevLett.59.381]

    Article  MathSciNet  Google Scholar 

  • Barthel, W., Hartmann, A.K., Weigt, M., 2003. Solving satisfiability problems by fluctuations: the dynamics of stochastic local search algorithms. Phys. Rev. E, 67(6): 066104. [doi:10.1103/PhysRevE.67.066104]

    Article  Google Scholar 

  • Biere, A., Heule, M., Maaren, H., Walsh, T., 2009. Handbook of Satisfiability. IOS Press, the Netherlands.

    MATH  Google Scholar 

  • Biroli, G., Cocco, S., Monasson, R., 2002. Phase transitions and complexity in computer science: an overview of the statistical physics approach to the random satisfiability problem. Phys. A, 306:381–394. [doi:10.1016/S0378-4371 (02)00516-2]

    Article  MATH  Google Scholar 

  • Boettcher, S., Percus, A.G., 2000. Nature’s way of optimizing. Artif. Intell., 119(1–2):275–286. [doi:10.1016/S0004-3702 (00)00007-2]

    Article  MATH  Google Scholar 

  • Boettcher, S., Percus, A.G., 2001a. Optimization with extremal dynamics. Phys. Rev. Lett., 86(23):5211–5214. [doi:10.1103/PhysRevLett.86.5211]

    Article  Google Scholar 

  • Boettcher, S., Percus, A.G., 2001b. Extremal optimization for graph partitioning. Phys. Rev. E, 64(2):026114. [doi:10.1103/PhysRevE.64.026114]

    Article  Google Scholar 

  • Boettcher, S., Percus, A.G., 2004. Extremal optimization at the phase transition for the three-coloring problem. Phys. Rev. E, 69(6):066703. [doi:10.1103/PhysRevE.69.066703]

    Article  Google Scholar 

  • Cheeseman, P., Kanefsky, B., Taylor, W.M., 1991. Where the Really Hard Problems Are. Proc. 12th Int. Joint Conf. on Artificial Intelligence, p.331–337.

  • Garey, M.R., Johnson, D.S., 1979. Computers and Intractability: a Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York.

    MATH  Google Scholar 

  • Glover, F., 1989. Tabu search: part I. ORSA J. Comp., 1(3): 190–206. [doi:10.1287/ijoc.1.3.190]

    MATH  Google Scholar 

  • Gomes, C., Selman, B., 2002. Satisfied with physics. Science, 297(5582):784–785. [doi:10.1126/science.1074599]

    Article  Google Scholar 

  • Haken, H., Wolf, H.C., 1996. The Physics of Atoms and Quanta (5th Ed.). Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Hamacher, K., 2007. Adaptive extremal optimization by detrended fluctuation analysis. J. Comput. Phys., 227(2): 1500–1509. [doi:10.1016/j.jcp.2007.09.013]

    Article  MATH  Google Scholar 

  • Hansen, P., Jaumard, B., 1990. Algorithms for maximum satisfiability problems. Computing, 44(4):279–303. [doi:10.1007/BF02241270]

    Article  MATH  MathSciNet  Google Scholar 

  • Hartmann, A.K., Weigt, M., 2005. Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics. Wiley-VCH Verlag GmbH & Co. KGaA. [doi:10.1002/3527606734]

  • Heilmann, F., Hoffmann, K.H., Salamon, P., 2004. Best possible probability distribution over extremal optimization ranks. Europhys. Lett., 66(3):305–310. [doi:10.1209/epl/i2004-10011-3]

    Article  Google Scholar 

  • Hoffmann, K.H., Heilmann, F., Salamon, P., 2004. Fitness threshold accepting over extremal optimization ranks. Phys. Rev. E, 70(4):046704. [doi:10.1103/PhysRevE.70. 046704]

    Article  Google Scholar 

  • Hoos, H.H., Stuzle, T., 1999. SATLIB—the Satisfiability Library. Available from http://www.cs.ubc.ca/~hoos/SATLIB/benchm.html [Accessed on Mar. 18, 2010].

  • Kirkpatrick, S., Gelatt, C.D.Jr., Vecchi, M.P., 1983. Optimization by simulated annealing. Science, 220(4598):671–680. [doi:10.1126/science.220.4598.671]

    Article  MathSciNet  Google Scholar 

  • Menaï, M.B., Batouche, M., 2006. An effective heuristic algorithm for the maximum satisfiability problem. Appl. Intell., 24(3):227–239. [doi:10.1007/s10489-006-8514-7]

    Article  Google Scholar 

  • Mézard, M., Parisi, G., Zecchina, R., 2002. Analytic and algorithmic solution of random satisfiability problems. Science, 297(5582):812–815. [doi:10.1126/science.1073287]

    Article  Google Scholar 

  • Middleton, A.A., 2004. Improved extremal optimization for the Ising spin glass. Phys. Rev. E, 69(5):055701. [doi:10.1103/PhysRevE.69.055701]

    Article  MathSciNet  Google Scholar 

  • Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L., 1999. Determining computational complexity from characteristic ‘phase transition’. Nature, 400:133–137. [doi:10.1038/22055]

    Article  MathSciNet  Google Scholar 

  • Seitz, S., Alava, M., Orponen, P., 2005. Focused local search for random 3-satisfiability. J. Statist. Mech. Theor. Exp., 2005(6):P06006. [doi:10.1088/1742-5468/2005/06/P06006]

    Article  Google Scholar 

  • Selman, B., 2008. Computational science: a hard statistical view. Nature, 451(7179):639–640. [doi:10.1038/451639a]

    Article  Google Scholar 

  • Selman, B., Kautz, H.A., 1993. An Empirical Study of Greedy Local Search for Satisfiability Testing. Proc. 11th National Conf. on Artificial Intelligence, p.46–51.

  • Selman, B., Kautz, H.A., Cohen, B., 1994. Noise Strategies for Improving Local Search. Proc. 12th National Conf. on Artificial Intelligence, p.337–343.

  • Semerjian, G., Monasson, R., 2003. Relaxation and metastability in a local search procedure for random satisfiability problem. Phys. Rev. E, 67(6):066103. [doi:10.1103/PhysRevE.67.066103]

    Article  Google Scholar 

  • Szedmak, S., 2001. How to Find More Efficient Initial Solution for Searching. RUTCOR Research Report 49-2001, Rutgers Center for Operations Research, Rutgers University, Piscataway, New Jersey, USA.

    Google Scholar 

  • Zeng, G.Q., Lu, Y.Z., 2009. Survey on Computational Complexity with Phase Transitions and Extremal Optimization. Proc. 48th IEEE Conf. on Control and Decision & 28th Chinese Control Conf., p.4352–4359. [doi:10.1109/CDC.2009.5400085]

  • Zeng, G.Q., Lu, Y.Z., Mao, W.J., Chu, J., 2010. Study on probability distributions for evolution in modified extremal optimization. Phys. A, 389(9):1922–1930. [doi:10.1016/j.physa.2009.12.055]

    Article  Google Scholar 

  • Zhang, W.X., 2001. Phase transitions and backbones of 3-SAT and maximum 3-SAT. LNCS, 2239:153–167. [doi:10.1007/3-540-45578-7_11]

    Google Scholar 

  • Zhang, W.X., 2004. Configuration landscape analysis and backbone guided local search. Part I: satisfiability and maximum satisfiability. Artif. Intell., 158(1):1–26. [doi:10.1016/j.artint.2004.04.001]

    Article  MATH  Google Scholar 

  • Zhou, T., Bai, W.J., Cheng, L.L., Wang, B.H., 2005. Continuous extremal optimization for Lennard-Jones clusters. Phys. Rev. E, 72(1):016702. [doi:10.1103/PhysRevE.72.016702]

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Physics & Electronic Information Engineering, Wenzhou University, Wenzhou, 325035, China

    Guo-qiang Zeng

  2. State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, 310027, China

    Yong-zai Lu & Wei-Jie Mao

Authors
  1. Guo-qiang Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong-zai Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei-Jie Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-qiang Zeng.

Additional information

Project supported by the National Natural Science Foundation of China (No. 61074045), the National Basic Research Program (973) of China (No. 2007CB714000), and the National Creative Research Groups Science Foundation of China (No. 60721062)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zeng, Gq., Lu, Yz. & Mao, WJ. Modified extremal optimization for the hard maximum satisfiability problem. J. Zhejiang Univ. - Sci. C 12, 589–596 (2011). https://doi.org/10.1631/jzus.C1000313

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1631/jzus.C1000313

Key words

CLC number

Search

Navigation