Abstract
An important heuristic in local search algorithms for Satisfiability is focusing, i.e. restricting the selection of flipped variables to those appearing in presently unsatisfied clauses. We consider the behaviour on large randomly generated 3-SAT instances of two focused solution methods: WalkSAT and Focused Metropolis Search. The algorithms turn out to have qualitatively quite similar behaviour. Both are sensitive to the proper choice of their “noise” and “temperature” parameters, but with appropriately chosen values, both achieve solution times that scale linearly in the number of variables even for clauses-to-variables ratios α > 4.2. This is much closer to the satisfiability transition threshold α c ≈ 4.267 than has generally been assumed possible for local search algorithms.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aurell, E., Gordon, U., Kirkpatrick, S.: Comparing beliefs, surveys and random walks. In: NIPS 2004 (2004), arXiv.org: cond-mat/0406217
Barthel, W., Hartmann, A.K., Weigt, M.: Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms. Phys. Rev. E 67, 066104 (2003)
Braunstein, A., Mézard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability, arXiv.org:cs.CC/0212002
Dueck, G.: New optimization heuristics: the great deluge algorithm and the record-to-record travel. J. Comput. Phys. 104, 86–92 (1993)
Hoos, H.H.: An adaptive noise mechanism for WalkSAT. In: AAAI 2002, pp. 655–660 (2002)
Hoos, H.H., Stützle, T.: Towards a characterisation of the behaviour of stochastic local search algorithms for SAT. Artif. Intell. 112, 213–232 (1999)
Jia, H., Moore, C., Selman, B.: From spin glasses to hard satisfiable formulas. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 199–210. Springer, Heidelberg (2005)
McAllester, D., Selman, B., Kautz, H.: Evidence for invariants in local search. In: AAAI 1997, pp. 321–326 (1997)
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)
Mitchell, D., Selman, B., Levesque, H.: Hard and easy distributions of SAT problems. In: AAAI 1992, pp. 459–465 (1992)
Papadimitriou, C.: On selecting a satisfying truth assignment. In: FOCS 1991, pp. 163–169 (1991)
Parisi, G.: On local equilibrium equations for clustering states, arXiv.org:cs.CC/0212047
Parkes, A.J.: Distributed local search, phase transitions, and polylog time. In: Proc. Workshop on Stochastic Search Algorithms, IJCAI 2001 (2001)
Parkes, A.J.: Scaling properties of pure random walk on random 3-SAT. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 708–713. Springer, Heidelberg (2002)
Seitz, S., Orponen, P.: An efficient local search method for random 3-satisfiability. In: Workshop on Typical Case Complexity and Phase Transitions, LICS 2003. Electronic Notes in Discrete Mathematics, vol. 16. Elsevier, Amsterdam (2003)
Seitz, S., Alava, M., Orponen, P.: Focused local search for random 3-satisfiability, arXiv.org:cond-mat/0501707
Selman, B., Kautz, H., Cohen, B.: Local search strategies for satisfiability testing. In: Cliques, Coloring, and Satisfiability. AMS DIMACS Series, vol. 26, pp. 521–532 (1996)
Semerjian, G., Monasson, R.: Relaxation and metastability in a local search procedure for the random satisfiability problem. Phys. Rev. E 67, 066103 (2003)
Semerjian, G., Monasson, R.: A study of Pure Random Walk on random satisfiability problems with “physical” methods. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 120–134. Springer, Heidelberg (2004)
Wei, W., Selman, B.: Accelerating random walks. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 216–232. Springer, Heidelberg (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Seitz, S., Alava, M., Orponen, P. (2005). Threshold Behaviour of WalkSAT and Focused Metropolis Search on Random 3-Satisfiability. In: Bacchus, F., Walsh, T. (eds) Theory and Applications of Satisfiability Testing. SAT 2005. Lecture Notes in Computer Science, vol 3569. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11499107_41
Download citation
DOI: https://doi.org/10.1007/11499107_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26276-3
Online ISBN: 978-3-540-31679-4
eBook Packages: Computer ScienceComputer Science (R0)