Limit Theorems for Random MAX-2-XORSAT

  • Vonjy Rasendrahasina
  • Vlady Ravelomanana
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)


We consider random instances of the MAX-2-XORSAT optimization problem. A 2-XOR formula is a conjunction of Boolean equations (or clauses) of the form x ⊕ y = 0 or x ⊕ y = 1. The MAX-2-XORSAT problem asks for the maximum number of clauses which can be satisfied by any assignment of the variables in a 2-XOR formula. In this work, formula of size m on n Boolean variables are chosen uniformly at random from among all \(\binom{n(n-1) }{ m}\) possible choices. Denote by X n,m the minimum number of clauses that can not be satisfied in a formula with n variables and m clauses. We give precise characterizations of the r.v. X n,m around the critical density \(\frac{m}{n} \sim \frac{1}{2}\) of random 2-XOR formula. We prove that for random formulas with m clauses X n,m converges to a Poisson r.v. with mean \(-\frac{1}{4}\log(1-2c)-\frac{c}{2}\) when m = cn, c ∈ ]0,1/2[ constant. If \(m= \frac{n}{2}-\frac{\mu}{2}n^{2/3}\), μ and n are both large but μ = o(n 1/3), \(\frac{X_{n,m}-\lambda} {\sqrt{\lambda}}\) with \(\lambda=\frac{\log{n}}{12} -\frac{\log{\mu}}{4}\) is normal. If \(m = \frac{n}{2} + O(1)n^{2/3}\), \(\frac{X_{n,m}- \frac{\log{n}}{12}}{\sqrt{\frac{\log{n}}{12}}}\) is normal. If \(m = \frac{n}{2} + \frac{\mu}{2}n^{2/3}\) with 1 ≪ μ = o(n 1/3) then \(\frac{ 12X_{n,m}}{2\mu^3+\log{n}-3\log(\mu)} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1\). For any absolute constant ε> 0, if μ = εn 1/3 then \(\frac{8(1+\varepsilon)}{n( \varepsilon^2 - \sigma^2)} X_{n,m} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1\) where σ ∈ (0,1) is the solution of (1 + ε)e  − ε  = (1 − σ)e σ . Thus, our findings describe phase transitions in the optimization context similar to those encountered in decision problems.


MAX XORSAT Constraint Satisfaction Problem Phase transition Random graph Analytic Combinatorics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Achlioptas, D., Moore, C.: Random k-SAT: Two moments suffice to cross a sharp threshold. SIAM Journal of Computing 36(3), 740–762 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Achlioptas, D., Naor, A., Peres, Y.: On the maximum satisfiability of random formulas. Journal of the ACM 54(2) (2007)Google Scholar
  3. 3.
    Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-like Structures. Cambridge University Press, Cambridge (1997)Google Scholar
  4. 4.
    Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics (1985)Google Scholar
  5. 5.
    Bollobás, B., Borgs, C., Chayes, J.T., Kim, J.H., Wilson, D.B.: The scaling window of the 2-SAT transition. Random Structures and Algorithms 18, 201–256 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Coppersmith, D., Hajiaghayi, M.T., Gamarnik, D., Sorkin, G.B.: Random MAX-SAT, random MAX-CUT, and their phase transitions. Random Structures and Algorithms 24(4), 502–545 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Creignou, N., Daudé, H.: Satisfiability threshold for random XOR-CNF formula. Discrete Applied Mathematics 96-97(1-3), 41–53 (1999)Google Scholar
  8. 8.
    Creignou, N., Daudé, H.: Coarse and sharp thresholds for random k-XOR-CNF satisfiability. Theoretical Informatics and Applications 37(2), 127–147 (2003)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Creignou, N., Daudé, H.: Coarse and sharp transitions for random generalized satisfiability problems. In: Proc. of the third Colloquium on Mathematics and Computer Science, pp. 507–516. Birkhäuser, Basel (2004)Google Scholar
  10. 10.
    Daudé, H., Ravelomanana, V.: Random 2-XORSAT phase transition. Algorithmica (2009); In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 12–23. Springer, Heidelberg (2008) Google Scholar
  11. 11.
    de Bruin, N.G.: Asymptotic Methods in Analysis. Dover, New York (1981)Google Scholar
  12. 12.
    DeLaurentis, J.: Appearance of complex components in a random bigraph. Random Structures and Algorithms 7(4), 311–335 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Dubois, O., Mandler, J.: The 3-XOR-SAT threshold. In: Proceedings of the 43th Annual IEEE Symposium on Foundations of Computer Science, pp. 769–778 (2002)Google Scholar
  15. 15.
    Dubois, O., Monasson, R., Selman, B., Zecchina, R.: Phase transitions in combinatorial problems. Theoretical Computer Science 265(1-2) (2001)Google Scholar
  16. 16.
    Flajolet, P., Knuth, D.E., Pittel, B.: The first cycles in an evolving graph. Discrete Mathematics 75(1-3), 167–215 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  18. 18.
    Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. Journal of Statistical Physics 111, 535–564 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Friedgut, E.: Sharp thresholds of graph properties, and the k-SAT problem. appendix by J. Bourgain. Journal of the A.M.S. 12(4), 1017–1054 (1999)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Goerdt, A.: A sharp threshold for unsatisfiability. Journal of Computer and System Sciences 53(3), 469–486 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Goulden, I.P., Jackson, D.M.: Combinatorial enumeration. John Wiley and Sons, Chichester (1983)zbMATHGoogle Scholar
  22. 22.
    Harary, F., Palmer, E.: Graphical enumeration. Academic Press, New-York (1973)zbMATHGoogle Scholar
  23. 23.
    Harary, F., Uhlenbeck, G.: On the number of Husimi trees, I. Proceedings of the National Academy of Sciences 39, 315–322 (1953)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Janson, S., Knuth, D.E., Łuczak, T., Pittel, B.: The birth of the giant component. Random Structures and Algorithms 4(3), 233–358 (1993)zbMATHCrossRefGoogle Scholar
  25. 25.
    Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley-Interscience, Hoboken (2000)zbMATHGoogle Scholar
  26. 26.
    Kolchin, V.F.: Random graphs. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  27. 27.
    Pittel, B., Wormald, N.C.: Counting connected graphs inside-out. Journal of Comb. Theory, Ser. B 93(2), 127–172 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ravelomanana, V.: Another proof of Wright’s inequalities. Information Processing Letters 104(1), 36–39 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Wright, E.M.: The number of connected sparsely edged graphs. Journal of Graph Theory 1, 317–330 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Wright, E.M.: The number of connected sparsely edged graphs III: Asymptotic results. Journal of Graph Theory 4(4), 393–407 (1980)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Vonjy Rasendrahasina
    • 1
  • Vlady Ravelomanana
    • 2
  1. 1.LIPN - UMR CNRS 7030Université de Paris Nord.VilletaneuseFrance
  2. 2.LIAFA - UMR CNRS 7089Université Denis Diderot.Paris Cedex 13France

Personalised recommendations