Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of interactions is induced by a sparse random graph or hypergraph. One example of such a model is the graph coloring problem on the Erdős–Renyi random graph G(n, d n), which can be viewed as the zero temperature case of the Potts antiferromagnet. The cavity method predicts that in addition to the k-colorability phase transition studied intensively in combinatorics, there exists a second phase transition called the condensation phase transition ( in Proc Natl Acad Sci 104:10318–10323, 2007). In fact, there is a conjecture as to the precise location of this phase transition in terms of a certain distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k0.
Perhaps the most plausible stab at understanding Belief Propagation Guided Decimation is the non-rigorous contribution @cite_10 . Roughly speaking, the result of the Belief Propagation fixed point iteration after @math iterations can be expected to yield a good approximation to the actual marginal distribution iff there is no condensation among the remaining list colorings. If so, one should expect that the algorithm actually finds a @math -coloring if condensation does not occur at any step @math . Thus, we look at a two-dimensional phase diagram'' parametrised by the average degree @math and the time @math . We need to identify the line that marks the (suitably defined) condensation phase transition in this diagram. deals with the case @math , and it would be most interesting to see if the present techniques extend to @math . Attempts at (rigorously) analysing message passing algorithms along these lines have been made for random @math -SAT, but the current results are far from precise @cite_4 @cite_16 .
@cite_16 @cite_10 @cite_4
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We study the antiferromagnetic Potts model on the Poissonian Erdős-Renyi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades) and from an entropy positivity argument. We introduce a version of the cavity method for diluted mean-field spin models that allows the computation of thermodynamic quantities similar to the Franz–Parisi quenched potential in sparse random graph models. This method is developed in the particular case of partially decimated random constraint satisfaction problems. This allows us to develop a theoretical understanding of a class of algorithms for solving constraint satisfaction problems, in which elementary degrees of freedom are sequentially assigned according to the results of a message passing procedure (belief propagation). We confront this theoretical analysis with the results of extensive numerical simulations. 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 (Mezard, Parisi, Zecchina: Science 2002; Braunstein, Mezard, 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.