Efficient 2 and 3-Flip Neighborhood Search Algorithms for the MAX SAT

  • Mutsunori Yagiura
  • Toshihide Ibaraki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)


For problems SAT and MAX SAT, local search algorithms are widely acknowledged as one of the most effective approaches. Most of the local search algorithms are based on the 1-flip neighborhood, which is the set of solutions obtainable by flipping the truth assignment of one variable. In this paper, we consider r-flip neighborhoods for r ≥ 2, and propose, for r = 2, 3, new implementations that reduce the number of candidates in the neighborhood without sacrificing the solution quality. For 2-flip (resp., 3-flip) neighborhood, we show that its expected size is O(n + m) (resp., O(m + t 2 n)), which is usually much smaller than the original size O(n 2) (resp., O(n 3)), where n is the number of variables, m is the number of clauses and t is the maximum number of appearances of one variable. Computational results tell that these estimates by the expectation well represent the real performance. These neighborhoods are then used under the framework of tabu search etc., and compared with other existing algorithms based on 1-flip neighborhood. The results exhibit good prospects of the proposed algorithms.


Local Search Tabu Search Local Search Algorithm Satisfying Assignment Iterate Local Search 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    B. Cha, K. Iwama, Y. Kambayashi and S. Miyazaki, Local search algorithms for partial MAXSAT, Proc. AAAI (1997) 263–268.Google Scholar
  2. 2.
    J.W. Freeman, Improvements to propositional satisfiability search algorithms, Dissertation, University of Pennsylvania, 1995.Google Scholar
  3. 3.
    A.S. Fukunaga, Variable-selection heuristics in local search for SAT, Proc. AAAI (1997) 275–280.Google Scholar
  4. 4.
    J. Gu, Efficient local search for very large-scale satisfiability problems, SIGART Bulletin, 3 (1992) 8–12.CrossRefGoogle Scholar
  5. 5.
    P. Hansen and B. Jaumard, Algorithms for the maximum satisfiability problem, Computing, 44 (1990) 279–303.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J.N. Hooker and C. Fedjki, Branch-and-cut solution of inference problems in propositional logic, Annals of Mathematics and Artificial Intelligence, 1 (1990) 123–139.zbMATHCrossRefGoogle Scholar
  7. 7.
    B. Mazure, L. Saïs and É. Grégoire, Tabu search for SAT, Proc. AAAI (1997) 281–285.Google Scholar
  8. 8.
    K. Nonobe and T. Ibaraki, A tabu search approach to the CSP (constraint satisfaction problem) as a general problem solver, European J. Oper. Res., Special Issue on Tabu Search (to appear).Google Scholar
  9. 9.
    P.M. Pardalos, L.S. Pitsoulis and M.G.C. Resende, A parallel GRASP for MAXSAT problems, LNCS, 1180 (1996) 575–585.Google Scholar
  10. 10.
    M.G.C. Resende, L.S. Pitsoulis and P.M. Pardalos, Approximate solution of weighted MAX-SAT problems using GRASP, DIMACS Series on Discrete Mathematics and Theoretical Computer Science (to appear).Google Scholar
  11. 11.
    B. Selman, H. Levesque and D. Mitchell, A new method for solving hard satisfiability problems, Proc. AAAI (1992) 440–446.Google Scholar
  12. 12.
    B. Selman and H.A. Kautz, An empirical study of greedy local search for satisfiability testing, Proc. AAAI (1993) 46–51.Google Scholar
  13. 13.
    B. Selman and H.A. Kautz, Domain-independent extensions to GSAT: solving large structured satisfiability problems, Proc. IJCAI (1993) 290–295.Google Scholar
  14. 14.
    B. Selman, H.A. Kautz and B. Cohen, Noise strategies for improving local search, Proc. AAAI (1994) 337–343.Google Scholar
  15. 15.
    B. Selman, H.A. Kautz and B. Cohen, Local search strategies for satisfiability testing, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 26 (1996) 521–531.Google Scholar
  16. 16.
    M. Yagiura and T. Ibaraki, Efficient 2 and 3-flip neighborhood search algorithms for the MAX SAT — Part I: Theoretical analysis, prepared for publication; Part II: Experimental results, ditto.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Mutsunori Yagiura
    • 1
  • Toshihide Ibaraki
    • 1
  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

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