Skip to main content
SpringerLink
Log in

The analysis of expected fitness and success ratio of two heuristic optimizations on two bimodal MaxSAT problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Heuristic algorithms, especially hill-climbing algorithms, are prone to being trapped by local optimization. Many researchers have focused on analyzing convergence and runtime of some heuristic algorithms on SAT-solving problems. However, there is rare work on analyzing success ratio (the ratio of the number of runs that find the global maxima of optimization versus the total runs) and expected fitness at each generation. Expected fitness at each generation could lead us to better understand the expected fitness of a heuristic algorithm could find on the problem to be solve at a certain generation. Success ratio help us understand with what a probability a heuristic algorithm could find the global optimization of the problem to be solved. So, this paper analyzed expected fitness and success ratio of two hill-climbing algorithms on two bimodal MaxSAT problems by using Markov chain. The theoretical and experimental results showed that though hill-climbing algorithms (both hill-climbing RandomWalk and Local (1+1)EA) converged faster, they could not always find global maxima of bimodal SAT-solving problems. The success ratios of hill-climbing algorithms usually have their own limits. The limit of success ratio is dependent on the SAT-solving problem itself and the expected distribution of initial bit string, and independent on whether hill-climbing RandomWalk or Local (1+1)EA is used.

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

  1. Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing 1971, Shaker Heights, Ohio, United States (1971)

  2. Zhou Y., He J., Nie Q.: A comparative runtime analysis of heuristic algorithms for satisfiability problems. Artif. Intell. 173, 240–257 (2009)

    Article  Google Scholar 

  3. Kusper G., Csõke L., Kovásznai G.: Simplifying the propositional satisfiability problem by sub-model propagation. Annales Mathematicae et Informaticae 35, 75–94 (2008)

    Google Scholar 

  4. Garey M.R., Johnson D.S.: Computers and Intractabilitya Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    Google Scholar 

  5. Lardeux F., Saubion F., Hao J.K.: GASAT: a genetic local search algorithm for the satisfiability problem. Evol. Comput. 14, 223–253 (2006)

    Article  Google Scholar 

  6. Gottlieb J., Marchiori E., Rossi C.: Evolutionary algorithms for the satisfiability problem. Evol. Comput. 10, 35–50 (2002)

    Article  Google Scholar 

  7. Alekhnovich M., Ben-Sasson E.: Linear upper bounds for random walk on small density random 3-CNF. SIAM J. Comput. 36, 1248–1263 (2006)

    Article  Google Scholar 

  8. Coja-Oghlan A, Feige U, Frieze A, Krivelevich M, et al.: On smoothed k-CNF formulas and the Walksat algorithm. In: Proceedings of 20th SODA, New York (2009)

  9. Hirsch E.A., Kojevnikov A.: UnitWalk: A new SAT solver that uses local search guided by unit clause elimination. Ann. Math. Artif. Intell. 43, 91–111 (2005)

    Google Scholar 

  10. Kirkpatrick S., Gelatt C D., Vecchi M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  Google Scholar 

  11. Semerjian, G., Monasson, R.: A Study of pure random walk on random satisfiability problems with “physical” methods. In: Giunchiglia, E., Tacchella, A. (eds.) Proceedings of the SAT 2003 conference, Lecture Notes in Computer Science, vol. 2919, pp. 120–134 (2004)

  12. Sasaki G.H., Hajek B.: The time complexity of maximum matching by simulated annealing. J. ACM 35, 387–403 (1988)

    Article  Google Scholar 

  13. Garnier J., Kallel L.: Rigorous hitting times for binary mutations. Evol. Comput. 7, 173–203 (1999)

    Article  Google Scholar 

  14. Krishnamachari B., Xie X., Selman B. et al.: Analysis of random noise and RandomWalk algorithms. Lecture Notes in Computer Science 1894, 278–290 (2000)

    Article  Google Scholar 

  15. He J., Yao X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127, 57–85 (2001)

    Article  Google Scholar 

  16. Droste S., Jansen T., Wegener I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)

    Article  Google Scholar 

  17. He J., Yao X.: Towards an analytic framework for analysing the computation time of evolutionary algorithms. Artif. Intell. 145, 59–97 (2003)

    Article  Google Scholar 

  18. Oliveto P.S., He J., Yao X.: Time complexity of evolutionary algorithms for combinatorial optimization: a decade of results. Int. J. Autom. Comput. 4, 281–293 (2007)

    Article  Google Scholar 

  19. Yu Y., Zhou Z.H.: A new approach to estimating the expected first hitting time of evolutionary algorithms. Artif. Intell. 172, 1809–1832 (2008)

    Article  Google Scholar 

  20. Mahajan M., Raman V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31, 335–354 (1999)

    Article  Google Scholar 

  21. Du, D.Z., Gu, J., Pardalos, P.M. (eds.): Satisfiability Problem: Theory and Applications. DIMACS Series vol. 35, American Mathematical Society (1998)

  22. Resende M.G.C., Pitsoulis L.S., Pardalos P.M.: FORTRAN subroutines for computing approximate solutions of weighted MAX-SAT problems using GRASP. Discret. Appl. Math. 100, 95–113 (1999)

    Article  Google Scholar 

  23. Festa, P., Pardalos, P.M., Pitsoulis, L.S. et al.: GRASP with path relinking for the weighted MAXSAT problem. ACM J. Exp. Algorithmics. 11, 1–16, Article No. 2.4 (2006)

    Google Scholar 

  24. Hobert J.P., Tan A., Liu R.: When is Eaton’s Markov chain irreducible?. Bernoulli 13, 641–652 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, South China University of Technology, Guangzhou, 510006, China

    Xinsheng Lai & Yuren Zhou

  2. School of Mathematics and Computer, ShangRao Normal University, Shangrao, 334001, China

    Xinsheng Lai

Authors
  1. Xinsheng Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuren Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinsheng Lai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lai, X., Zhou, Y. The analysis of expected fitness and success ratio of two heuristic optimizations on two bimodal MaxSAT problems. J Glob Optim 54, 745–764 (2012). https://doi.org/10.1007/s10898-011-9790-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9790-2

Keywords

Search

Navigation