Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. We present a polynomial time algorithm that finds a satisfying assignment of @math with high probability for constraint densities @math , where @math . Previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond @math [A. Frieze and S. Suen, J. Algorithms, 20 (1996), pp. 312-355].
Survey Propagation is a modification of Belief Propagation that aims to approximate the marginal probabilities induced by a particular (non-uniform) probability distribution on the set of satisfying assignments @cite_11 . It can be combined with a decimation procedure as well to obtain a heuristic for a satisfying assignment. There is (non-rigorous) evidence that for most of the satisfiable regime (actually @math ) Belief and Survey Propagation are essentially equivalent @cite_15 . Hence, there is no evidence that Survey Propagation finds satisfying assignments beyond @math for general @math .
@cite_15 @cite_11
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Survey Propagation is an algorithm designed for solving typical instances of random constraint satisfiability problems. It has been successfully tested on random 3-SAT and random @math graph 3-coloring, in the hard region of the parameter space. Here we provide a generic formalism which applies to a wide class of discrete Constraint Satisfaction Problems. Belief propagation is a remarkably effective tool for inference, even when applied to networks with cycles. It may be viewed as a way to seek the minimum of the Bethe free energy, though with no convergence guarantee in general. A variational perspective shows that, compared to exact inference, this minimization employs two forms of approximation: (i) the true entropy is approximated by the Bethe entropy, and (ii) the minimization is performed over a relaxation of the marginal polytope termed the local polytope. Here we explore when and how the Bethe approximation can fail for binary pairwise models by examining each aspect of the approximation, deriving results both analytically and with new experimental methods. Saliency propagation has been widely adopted for identifying the most attractive object in an image. The propagation sequence generated by existing saliency detection methods is governed by the spatial relationships of image regions, i.e., the saliency value is transmitted between two adjacent regions. However, for the inhomogeneous difficult adjacent regions, such a sequence may incur wrong propagations. In this paper, we attempt to manipulate the propagation sequence for optimizing the propagation quality. Intuitively, we postpone the propagations to difficult regions and meanwhile advance the propagations to less ambiguous simple regions. Inspired by the theoretical results in educational psychology, a novel propagation algorithm employing the teaching-to-learn and learning-to-teach strategies is proposed to explicitly improve the propagation quality. In the teaching-to-learn step, a teacher is designed to arrange the regions from simple to difficult and then assign the simplest regions to the learner. In the learning-to-teach step, the learner delivers its learning confidence to the teacher to assist the teacher to choose the subsequent simple regions. Due to the interactions between the teacher and learner, the uncertainty of original difficult regions is gradually reduced, yielding manifest salient objects with optimized background suppression. Extensive experimental results on benchmark saliency datasets demonstrate the superiority of the proposed algorithm over twelve representative saliency detectors. 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.