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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)

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics… (More)

—For every í µí¼ > 0, and integer í µí± ≥ 3, we show that given an í µí±-vertex graph that has an induced í µí±-colorable subgraph of size (1−í µí¼)í µí± , it is NP-hard to find an independent set of size í µí± í µí± 2 .

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)

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)

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)

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)

We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.