On belief propagation guided decimation for random k-SAT

  • Authors:
  • Amin Coja-Oghlan

  • Affiliations:
  • University of Warwick, UK

  • Venue:
  • Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2011

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Abstract

Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that Φ is satisfiable for clause/variable ratios m/n ≤ rk ~ 2k ln 2 with high probability (Achlioptas, Moore: SICOMP 2006; Achlioptas, Peres: J. AMS 2004). Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m/n ~ rk · ln(k)/k with a non-vanishing probability. In fact, the density m/n ~ rk · ln(k)/k seems to form a barrier for a broad class of local search algorithms (Achlioptas, Coja-Oghlan: FOCS 2008). On the basis of deep but non-rigorous statistical mechanics considerations, a message passing algorithm called belief propagation guided decimation for solving random k-SAT has been forward (Mézard, Parisi, Zecchina: Science 2002; Braunstein, Mézard, Zecchina: RSA 2005). Experiments suggest that the algorithm might succeed for densities very close to rk for k = 3, 4, 5 (Kroc, Sabharwal, Selman: SAC 2009). Furnishing the first rigorous analysis of belief propagation guided decimation on random k-SAT, the present paper shows that the algorithm fails to find a satisfying assignment already for m/n ≥ ρ · rk/k, for a constant ρ 0 independent of k.