Advertisement

Spines of random constraint satisfaction problems: definition and connection with computational complexity

  • Gabriel IstrateEmail author
  • Stefan Boettcher
  • Allon G. Percus
Article

Abstract

We study the connection between the order of phase transitions in combinatorial problems and the complexity of decision algorithms for such problems. We rigorously show that, for a class of random constraint satisfaction problems, a limited connection between the two phenomena indeed exists. Specifically, we extend the definition of the spine order parameter of Bollobás et al. [10] to random constraint satisfaction problems, rigorously showing that for such problems a discontinuity of the spine is associated with a 2Ω(n) resolution complexity (and thus a 2Ω(n) complexity of DPLL algorithms) on random instances. The two phenomena have a common underlying cause: the emergence of “large” (linear size) minimally unsatisfiable subformulas of a random formula at the satisfiability phase transition.

We present several further results that add weight to the intuition that random constraint satisfaction problems with a sharp threshold and a continuous spine are “qualitatively similar to random 2-SAT”. Finally, we argue that it is the spine rather than the backbone parameter whose continuity has implications for the decision complexity of combinatorial problems, and we provide experimental evidence that the two parameters can behave in a different manner.

Keywords

constraint satisfaction problems phase transitions spine resolution complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Achlioptas, A. Chtcherba, G. Istrate and C. Moore, The phase transition in random 1-in-k SAT and NAE 3SAT, in: Proc. of the 13th ACM-SIAM Symposium on Discrete Algorithms (2001). Journal version in preparation. Google Scholar
  2. [2]
    D. Achlioptas and E. Friedgut, A sharp threshold for k-colorability, Random Structures and Algorithms 14(1) (1999) 63–70. CrossRefGoogle Scholar
  3. [3]
    P. Beame, R. Karp, T. Pitassi and M. Saks, The efficiency of resolution and Davis–Putnam procedures, SIAM Journal of Computing 31(4) (2002) 1048–1075. CrossRefGoogle Scholar
  4. [4]
    P. Beame and T. Pitassi, Propositional proof complexity: Past present and future, in: Current Trends in Theoretical Computer Science (2001) pp. 42–70. Google Scholar
  5. [5]
    E. Ben-Sasson and A. Wigderson, Short proofs are narrow: Resolution made simple, Journal of the ACM 48(2) (2001). Google Scholar
  6. [6]
    S. Boettcher, A extremal optimization of graph partition at the percolation threshold, J. Phys. Math. Gen. 32 (1999) 5201–5211. CrossRefGoogle Scholar
  7. [7]
    S. Boettcher and A. Percus, Nature’s way of optimizing, Artificial Intelligence 119 (2000) 275–286. CrossRefGoogle Scholar
  8. [8]
    S. Boettcher and A.G. Percus, Extremal optimization at the phase transition of the 3-coloring problem, Physical Review E 69 (2004) 066703. CrossRefGoogle Scholar
  9. [9]
    B. Bollobás, Random Graphs (Academic Press, New York, 1985). Google Scholar
  10. [10]
    B. Bollobás, C. Borgs, J.T. Chayes, J.H. Kim and D.B. Wilson, The scaling window of the 2-SAT transition, Technical Report, Los Alamos e-print server, http://xxx.lanl.gov/ps/math.CO/9909031, 1999.
  11. [11]
    V. Chvátal and E. Szemerédi, Many hard examples for resolution, Journal of the ACM 35(4) (1988) 759–768. CrossRefGoogle Scholar
  12. [12]
    N. Creignou and H. Daudé, Combinatorial sharpness criterion and phase transition classification for random CSPs, Information and Computation 190(2) (2004) 220–238. CrossRefGoogle Scholar
  13. [13]
    J. Culberson and I. Gent, Frozen development in graph coloring, Theoretical Computer Science 265(1/2) (2001) 227–264. CrossRefGoogle Scholar
  14. [14]
    E. Friedgut, Necessary and sufficient conditions for sharp thresholds of graph properties, and the k-SAT problem with an appendix by J. Bourgain, Journal of the AMS 12 (1999) 1017–1054. Google Scholar
  15. [15]
    N. Immerman, Descriptive Complexity, Springer Graduate Texts in Computer Science (Springer, Berlin, 1999). Google Scholar
  16. [16]
    G. Istrate, Phase transitions and all that, Preprint CS.CC/0211012, ACM Computer Repository at arXiv.org. Google Scholar
  17. [17]
    G. Istrate, Threshold properties of random constraint satisfaction problems, accepted to a special volume of Discrete Applied Mathematics on typical-case complexity and phase transitions. Google Scholar
  18. [18]
    G. Istrate, Descriptive complexity and first-order phase transitions (in progress). Google Scholar
  19. [19]
    J. Krajicek, On the weak pigeonhole principle, Fundamenta Matematicae 170(1–3) (2001) 123–140. Google Scholar
  20. [20]
    O. Martin, R. Monasson and R. Zecchina, Statistical mechanics methods and phase transitions in combinatorial optimization problems, Theoretical Computer Science 265(1/2) (2001) 3–67. CrossRefGoogle Scholar
  21. [21]
    D. Mitchell, Resolution complexity of random constraints, in: Eigth International Conference on Principles and Practice of Constraint Programming (2002). Google Scholar
  22. [22]
    M. Molloy, Models for random constraint satisfaction problems, in: Proc. of the 32nd ACM Symposium on Theory of Computing (2002). Google Scholar
  23. [23]
    M. Molloy and M. Salavatipour, The resolution complexity of random constraint satisfaction problems, in: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science (2003). Google Scholar
  24. [24]
    R. Monasson and R. Zecchina, Statistical mechanics of the random k-SAT model, Physical Review E 56 (1997) 1357. CrossRefGoogle Scholar
  25. [25]
    R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman and L. Troyansky, Determining computational complexity from characteristic phase transitions, Nature 400(8) (1999) 133–137. CrossRefGoogle Scholar
  26. [26]
    R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman and L. Troyansky, 2+p-SAT: Relation of typical-case complexity to the nature of the phase transition, Random Structures and Algorithms 15(3/4) (1999) 414–435. CrossRefGoogle Scholar
  27. [27]
    M.A. Trick, color.c graph coloring code, available at http://mat.gsia.cmu.edu/COLOR/solvers/trick.c.
  28. [28]
    R. Mulet, A. Pagnani, M. Weigt and R. Zecchina, Coloring random graphs, Physical Review Letters 89 (2002) 268701. CrossRefPubMedGoogle Scholar
  29. [29]
    F. Ricci-Tersenghi, M. Weigt and R. Zecchina, Simplest random k-satisfiability problem, Physical Review E 63 (2001) 026702. CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Gabriel Istrate
    • 1
    Email author
  • Stefan Boettcher
    • 2
  • Allon G. Percus
    • 3
    • 4
  1. 1.CCS-5Los Alamos National LaboratoryLos AlamosUSA
  2. 2.Physics DepartmentEmory UniversityAtlantaUSA
  3. 3.CCS-3Los Alamos National LaboratoryLos AlamosUSA
  4. 4.UCLA Institute for Pure and Applied MathematicsLos AngelesUSA

Personalised recommendations