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Rigorous location of phase transitions in hard optimization problems

Nature volume 435, pages 759–764 (2005)Cite this article

Abstract

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme ‘hardness’, computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

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Figure 1: The Satisfiability Threshold Conjecture.
Figure 2: Our results for random Max k -SAT.
Figure 3
Figure 4: Plots of entropy, correlation, and their product for the vanilla second moment method.
Figure 5: Plots of the correlation function and its product with entropy for the weighted second moment method, when the weighing is given by equation (3).

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References

  1. Cook, S. A. The complexity of theorem-proving procedures. Proc. 3rd Ann. ACM Symp. on Theory of Computing 151–158 (1971)

  2. Cheeseman, P., Kanefsky, B. & Taylor, W. Where the really hard problems are. Proc. 12th Int. Joint Conf. on Artificial Intelligence 331–337 (1991)

  3. Selman, B., Levesque, H. & Mitchell, D. Hard and easy distributions of SAT problems. Proc. 10th Nat. Conf. on Artificial Intelligence 459–465 (1992)

  4. Kirkpatrick, S. & Selman, B. Critical behavior in the satisfiability of random boolean expressions. Science 264, 1297–1301 (1994)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  5. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B. & Troyansky, L. Determining computational complexity from characteristic ‘phase transitions’. Nature 400, 133–137 (1999)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  6. Fu, Y. & Anderson, P. W. Application of statistical mechanics to NP-complete problems in combinatorial optimisation. J. Phys. A 19, 1605–1620 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  7. Anderson, P. W. Solving problems in finite time. Nature 400, 115–116 (1999)

    Article  ADS  CAS  Google Scholar 

  8. Gomes, C. P. & Selman, B. Satisfied with physics. Science 297, 784–785 (2002)

    Article  CAS  PubMed  Google Scholar 

  9. Monasson, R. & Zecchina, R. Statistical mechanics of the random K-satisfiability model. Phys. Rev. E 56, 1357–1370 (1997)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  10. Mézard, M. & Zecchina, R. Random K-satisfiability: from an analytic solution to a new efficient algorithm. Phys. Rev. E 66, 056126 (2002)

    Article  ADS  Google Scholar 

  11. Mézard, M., Parisi, G. & Zecchina, R. Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)

    Article  ADS  PubMed  Google Scholar 

  12. Mertens, S., Mézard, M. & Zecchina, R. Threshold values of random k-SAT from the cavity method. To appear in Random Struct. Algorithms (in the press); preprint at http://arxiv.org/abs/cs.cc/0309020 (2003).

  13. Appel, K. & Haken, W. Every planar map is four colorable. Contemp. Math. 98, (1989)

  14. Achlioptas, D. & Moore, C. The asymptotic order of the random k-SAT threshold. Proc. 43rd Ann. IEEE Symp. on Foundations of Computer Science 779–788 (2002)

  15. Achlioptas, D. & Peres, Y. The threshold for random k-SAT is 2k log 2 - O(k). J. Am. Math. Soc. 17, 947–973 (2004)

    Article  Google Scholar 

  16. Friedgut, E. Sharp thresholds of graph properties, and the k-SAT problem. J. Am. Math. Soc. 12, 1017–1054 (1999)

    Article  MathSciNet  Google Scholar 

  17. Achlioptas, D., Naor, A. & Peres, Y. On the maximum satisfiability of random formulas. Proc. 44th Ann. IEEE Symp. on Foundations of Computer Science 362–370 (2003)

  18. Bollobás, B. Random Graphs Vol. 73, 2nd edn Cambridge Studies in Advanced Mathematics (Cambridge Univ. Press, Cambridge, 2001)

    Google Scholar 

  19. Achlioptas, D. & Naor, A. The two possible values of the chromatic number of a random graph. Proc. 36th Ann. ACM Symp. on Theory of Computing 587–593 (2004)

  20. Krzakala, F., Pagnani, A. & Weigt, M. Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs. Phys. Rev. E 70, 046705 (2004)

    Article  ADS  Google Scholar 

  21. Kirousis, L. M., Kranakis, E., Krizanc, D. & Stamatiou, Y. Approximating the unsatisfiability threshold of random formulas. Random Struct. Algorithms 12, 253–269 (1998)

    Article  MathSciNet  Google Scholar 

  22. Dubois, O., Boufkhad, Y. & Mandler, J. Typical random 3-SAT formulae and the satisfiability threshold. Proc. 11th Ann. ACM-SIAM Symp. on Discrete Algorithms 126–127 (2000)

  23. Kaporis, A. C., Kirousis, L. M. & Lalas, E. G. The probabilistic analysis of a greedy satisfiability algorithm. Proc. 10th Ann. European Symp. on Algorithms 574–585 (2002)

  24. Frieze, A. M. & Suen, S. Analysis of two simple heuristics on a random instance of k-SAT. J. Algorithms 20, 312–355 (1996)

    Article  MathSciNet  Google Scholar 

  25. Coppersmith, D., Gamarnik, D., Hajiaghayi, M. T. & Sorkin, G. B. Random MAX SAT random MAX CUT, and their phase transitions. Random Struct. Algorithms 24, 502–545 (2004)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank S. Kirkpatrick, J. Kleinberg and S. Mertens for feedback on the presentation of the results.

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Authors and Affiliations

  1. Microsoft Research, One Microsoft Way, Washington, 98052, Redmond, USA

    Dimitris Achlioptas & Assaf Naor

  2. Department of Statistics, University of California, California, 94720, Berkeley, USA

    Yuval Peres

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  1. Dimitris Achlioptas
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Correspondence to Dimitris Achlioptas.

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Achlioptas, D., Naor, A. & Peres, Y. Rigorous location of phase transitions in hard optimization problems. Nature 435, 759–764 (2005). https://doi.org/10.1038/nature03602

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