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Register a new userBelief Propagation on the random $k$-SAT model
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Added by
Noela M\\\"uller
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Corroborating a prediction from statistical physics, we prove that the Belief Propagation message passing algorithm approximates the partition function of the random $k$-SAT model well for all clause/variable densities and all inverse temperatures for which a modest absence of long-range correlations condition is satisfied. This condition is known as replica symmetry in physics language. From this result we deduce that a replica symmetry breaking phase transition occurs in the random $k$-SAT model at low temperature for clause/variable densities below but close to the satisfiability threshold.
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We show that throughout the satisfiable phase the normalised number of satisfying assignments of a random $2$-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to `true under a uniformly random satisfying assignment.
We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds to a distinct arrangement of transition probabilities, sometimes referred to as the pendulum arrangement. We reduce this problem to a sum over products, combinatorial optimization problem, proving that this unique structure always maximizes the escape time. This general property may influence studies in epidemiology, biology, and computer science to better understand escape time behavior and construct intruder-resilient networks.
We consider a natural model of inhomogeneous random graphs that extends the classical ErdH os-Renyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 1997]. In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic [EJP 1998] have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Levy-type processes. We, instead, look at the metric structure of these components and prove their Gromov-Hausdorff-Prokhorov convergence to a class of random compact measured metric spaces that have been introduced in a companion paper. Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general critical regime, and relies upon two key ingredients: an encoding of the graph by some Levy process as well as an embedding of its connected components into Galton-Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Levy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Levy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via regime-specific proofs, for instance: the case of ErdH os-Renyi random graphs obtained by Addario-Berry, Goldschmidt and B. [PTRF 2012], the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang [PTRF 2017], or the power-law case as considered by Bhamidi, Sen and van der Hofstad [PTRF 2018].
Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of these models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain survey propagation model associated to this problem. We believe that these methods may be applicable in a wide class of related problems.
One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT instances, arising in real-world applications. Various modifications have been suggested to define models of industrial SAT. Here, we focus mainly on the aspect of community structure. Namely, here the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Thus, we suggest a model of random industrial SAT, in which the central generalization with respect to random SAT is the additional community structure. There has been a lot of work on the satisfiability threshold of random $k$-SAT, starting with the calculation of the threshold of $2$-SAT, up to the recent result that the threshold exists for sufficiently large $k$. In this paper, we endeavor to study the satisfiability threshold for the proposed model of random industrial SAT. Our main result is that the threshold in this model tends to be smaller than its counterpart for random SAT. Moreover, under some conditions, this threshold even vanishes.
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