Journal of Combinatorial Optimization

, Volume 32, Issue 1, pp 51–66 | Cite as

Performances of pure random walk algorithms on constraint satisfaction problems with growing domains

  • Wei Xu
  • Fuzhou Gong


The performances of two types of pure random walk (PRW) algorithms for a model of constraint satisfaction problem with growing domains (called Model RB) are investigated. Threshold phenomenons appear for both algorithms. In particular, when the constraint density \(r\) is smaller than a threshold value \(r_d\), PRW algorithms can solve instances of Model RB efficiently, but when \(r\) is bigger than the \(r_d\), they fail. Using a physical method, we find out the threshold values for both algorithms. When the number of variables \(N\) is large, the threshold values tend to zero, so generally speaking PRW does not work on Model RB. By performing experiments, we show that PRW strategy cannot do better than other fundamental strategies.


Constraint satisfaction problems Model RB Random walk  Local search algorithms 



Partially supported by NSFC 61370052 and 61370156.


  1. Achlioptas D, Kirousis L, Kranakis E, Krizanc D, Molloy M, Stamatiou Y (1997) Random constraint satisfaction: a more accurate picture. In: Proceedings of CP. pp 107–120Google Scholar
  2. Alekhnovich M, Ben-Sasson E (2006) Linear upper bounds for random walk on small density random 3-cnfs. SIAM J Comput 36(5):1248–1263MathSciNetCrossRefMATHGoogle Scholar
  3. Alphonse E, Osmani A (2008) A model to study phase transition and plateaus in relational learning. In: Proceedings of of ILP. pp 6–23Google Scholar
  4. Barthel W, Hartmann AK, Weigt M (2003) Solving satisfiability problems by fluctuations: the dynamics of stochastic local search algorithms. Phys Rev E 67:066104CrossRefGoogle Scholar
  5. Broder AZ, Frieze AM, Upfal E (1993) On the satisfiability and maximum satisfiability of random 3-CNF formulas. In: Proceedings of SODA. pp 322–330Google Scholar
  6. Chao M, Franco J (1986) Probabilistic analysis of two heuristics for the 3-satisfiability problem. SIAM J Comput 15(4):1106–1118MathSciNetCrossRefMATHGoogle Scholar
  7. Coja-Oghlan A, Frieze A (2012) Analyzing Walksat on random formulas. In: Proceedings of ANALCO. pp 48–55Google Scholar
  8. Coja-Oghlan A, Feige U, Frieze A, Krivelevich M, Vilenchik D (2009) On smoothed \(k\)-CNF formulas and the Walksat algorithm. In: Proceedings of SODA. pp 451–460Google Scholar
  9. Fan Y, Shen J (2011) On the phase transitions of random k-constraint satisfaction problems. Artif Intell 175:914–927MathSciNetCrossRefMATHGoogle Scholar
  10. Fan Y, Shen J, Xu K (2012) A general model and thresholds for random constraint satisfaction problems. Artif Intell 193:1–17MathSciNetCrossRefMATHGoogle Scholar
  11. Gao Y, Culberson J (2007) Consistency and random constraint satisfaction problems. J Artif Intell Res 28:517–557MathSciNetMATHGoogle Scholar
  12. Gent I, Macintype E, Prosser P, Smith B, Walsh T (2001) Random constraint satisfaction: flaws and structure. Constraints 6(4):345–372MathSciNetCrossRefMATHGoogle Scholar
  13. Huang P, Yin MH (2014) An upper (lower) bound for Max (Min) CSP. Sci China Inf Sci 57:072109MathSciNetMATHGoogle Scholar
  14. Jiang W, Liu T, Ren T, Xu K (2011) Two hardness results on feedback vertex sets. In: Proceedings of FAW-AAIM. pp 233–243Google Scholar
  15. Kamath A, Motwani R, Palem K, Spirakis P (1995) Tail bounds for occupancy and the satisfiability threshold conjecture. Random Struct Algorithm 7:59–80MathSciNetCrossRefMATHGoogle Scholar
  16. Lecoutre C (2009) Constraint networks: techniques and algorithms. Wiley, HobokenCrossRefGoogle Scholar
  17. Liu T, Lin X, Wang C, Su K, Xu K (2011) Large hinge width on sparse random hypergraphs. In: Proceedings of IJCAI. pp 611–616Google Scholar
  18. Liu T, Wang C, Xu K (2014) Large hypertree width for sparse random hypergraphs. J Comb Optim. doi: 10.1007/s10878-013-9704-y
  19. Richter S, Helmert M, Gretton C (2007) A stochastic local search approach to vertex cover. In: Proceedings of KI. pp 412–426Google Scholar
  20. Rossi F, Van Beek P, Walsh T (eds) (2006) Handbook of constraint programming. Elsevier, AmsterdamMATHGoogle Scholar
  21. Schöning U (2002) A probabilistic algorithm for \(k\)-SAT based on limited local search and restart. Algorithmica 32:615–623MathSciNetCrossRefMATHGoogle Scholar
  22. Schöning U (1999) A probabilistic algorithm for \(k\)-SAT and constraint satisfaction problems. In: Proceedings of FOCS. pp 410–414Google Scholar
  23. Semerjian G, Monasson R (2004) A study of pure random walk on random satisfiability problems with physical methods. In: Proceedings of SAT. pp 120–134Google Scholar
  24. Semerjian G, Monasson R (2003) Relaxation and metastability in the random walk SAT search procedure. Phys Rev E 67:066103CrossRefGoogle Scholar
  25. Shen J, Ren Y (2014) Bounding the scaling window of random constraint satisfaction problems. J Comb Optim. doi: 10.1007/s10878-014-9789-y
  26. Smith BM (2001) Constructing an asymptotic phase transition in random binary constraint satisfaction problems. Theor Comput Sci 265:265–283MathSciNetCrossRefMATHGoogle Scholar
  27. Smith BM, Dyer ME (1996) Locating the phase transition in binary constraint satisfaction problems. Artif Intell 81:155–181MathSciNetCrossRefGoogle Scholar
  28. Wang C, Liu T, Cui P, Xu K (2011) A note on treewidth in random graphs. In: Proceedings of COCOA. pp 491–499Google Scholar
  29. Xu K, Li W (2000) Exact phase transitions in random constraint satisfaction problems. J Artif Intell Res 12:93–103MathSciNetMATHGoogle Scholar
  30. Xu K, Li W (2006) Many hard examples in exact phase transitions. Theor Comput Sci 355:291–302MathSciNetCrossRefMATHGoogle Scholar
  31. Xu K, Boussemart F, Hemery F, Lecoutre C (2007) Random constraint satisfaction: easy generation of hard (satisfiable) instances. Artif Intell 171:514–534MathSciNetCrossRefMATHGoogle Scholar
  32. Xu W (2014) An analysis of backtrack-free algorithm on a constraint satisfaction problem with growing domains (in Chineses). Acta Math Appl Sin (Chin Ser) 37(3):385–392MATHGoogle Scholar
  33. Zhao C, Zheng Z (2011) Threshold behaviors of a random constraint satisfaction problem with exact phase transitions. Inf Process Lett 111:985–988MathSciNetCrossRefMATHGoogle Scholar
  34. Zhao C, Zhang P, Zheng Z, Xu K (2012) Analytical and belief-propagation studies of random constraint satisfaction problems with growing domains. Phys Rev E 85:016106CrossRefGoogle Scholar
  35. Zhou G, Gao Z, Liu J (2014) On the constraint length of random k-CSP. J Comb Optim. doi: 10.1007/s10878-014-9731-3

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.Institute of Applied Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations