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The problem of identifying a planted assignment given a random k-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with O(n log n) clauses, there are distributions over clauses for which the best known efficient algorithms require n<sup>k/2</sup> clauses. We(More)
The Erd˝ os-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erd˝ os and Rényi states that the Erd˝ os-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this sem-inal work(More)
We present an algorithm for recovering planted solutions in two well-known models , the stochastic block model and planted constraint satisfaction problems (CSP), via a common generalization in terms of random bipartite graphs. Our algorithm matches up to a constant factor the best-known bounds for the number of edges (or constraints) needed for perfect(More)
1 We consider a bipartite stochastic block model on vertex sets V 1 and V 2 , with planted partitions in each, and ask at what densities efficient algorithms can recover the partition of the smaller vertex set. When |V 2 | |V 1 |, multiple thresholds emerge. We first locate a sharp threshold for detection of the partition, in the sense of the results of(More)
According to physics predictions, the free energy of random factor graph models that satisfy a certain " static replica symmetry " condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al., PNAS 2007]. Here we prove this conjecture for two general classes of random factor graph models, namely Poisson random factor graphs(More)
Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs. This general result implies the conjecture on the information-theoretic threshold in the disassortative sto-chastic block model [Decelle et al.: Phys. Rev. E(More)