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Reweighted belief propagation and quiet planting for random ksat. arXiv preprint arXiv:1203.5521
, 2012
"... We study the random Ksatisfiability problem using a partition function where each solution is reweighted according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighted partition function. This allows us to obtain several ..."
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We study the random Ksatisfiability problem using a partition function where each solution is reweighted according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighted partition function. This allows us to obtain several new results on the properties of random Ksatisfiability problem. In particular the reweighting allows to introduce a planted ensemble that generates instances that are, in some region of parameters, equivalent to random instances. We are hence able to generate at the same time a typical random SAT instance and one of its solutions. We study the relation between clustering and belief propagation fixed points and we give a direct evidence for the existence of purely entropic (rather than energetic) barriers between clusters in some region of parameters in the random Ksatisfiability problem. We exhibit, in some large planted instances, solutions with a nontrivial whitening core; such solutions were known to exist but were so far never found on very large instances. Finally, we discuss algorithmic hardness of such planted instances and we determine a region of parameters in which planting leads to satisfiable benchmarks that, up to our knowledge, are the hardest known.
SATISFIABILITY THRESHOLD FOR RANDOM REGULAR NAESAT
"... Abstract. We consider the random regular knaesat problem with n variables each appearing in exactly d clauses. For all k exceeding an absolute constant k0, we establish explicitly the satisfiability threshold d ‹ ” d‹pkq. We prove that for d ă d ‹ the problem is satisfiable with high probability ..."
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Abstract. We consider the random regular knaesat problem with n variables each appearing in exactly d clauses. For all k exceeding an absolute constant k0, we establish explicitly the satisfiability threshold d ‹ ” d‹pkq. We prove that for d ă d ‹ the problem is satisfiable with high probability while for d ą d ‹ the problem is unsatisfiable with high probability. If the threshold d ‹ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krzakała et al. (2007). Our proof verifies the onestep replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs. 1.
On the chromatic number of a random hypergraph
, 2012
"... We consider the problem of kcolouring a random runiform hypergraph with n vertices and cn edges, where k, r, c remain constant as n → ∞. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case r = 2, must have one of two easily computable values as n → ∞. W ..."
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We consider the problem of kcolouring a random runiform hypergraph with n vertices and cn edges, where k, r, c remain constant as n → ∞. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case r = 2, must have one of two easily computable values as n → ∞. We give a complete generalisation of this result to random uniform hypergraphs. 1
Frozen variables in random boolean constraint satisfaction problems
, 2012
"... We determine the exact freezing threshold, r f, for a family of models of random boolean constraint satisfaction problems, including NAESAT and hypergraph 2colouring, when the constraint size is sufficiently large. If the constraintdensity of a random CSP, F, in our family is greater than r f the ..."
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We determine the exact freezing threshold, r f, for a family of models of random boolean constraint satisfaction problems, including NAESAT and hypergraph 2colouring, when the constraint size is sufficiently large. If the constraintdensity of a random CSP, F, in our family is greater than r f then for almost every solution of F, a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations in which we change o(n) variables at a time, always switching to another solution. If the constraintdensity is less than r f, then almost every solution has o(n) frozen variables. Freezing is a key part of the clustering phenomenon that is hypothesized by nonrigorous techniques from statistical physics. The understanding of clustering has led to the development of advanced heuristics such as Survey Propogation. It has been suggested that the freezing threshold is a precise algorithmic barrier: There is reason to believe that for densities below r f the random CSPs can be solved using very simple algorithms, while for densities above r f one requires more sophisticated techniques in order to deal with frozen clusters. 0 1
The Gaussian RateDistortion Function of Sparse Regression Codes with Optimal Encoding”, arXiv preprint arXiv:1401.5272
, 2014
"... Abstract—We study the ratedistortion performance of Sparse Regression Codes where the codewords are linear combinations of subsets of columns of a design matrix. It is shown that with minimumdistance encoding and squared error distortion, these codes achieve R∗(D), the Shannon ratedistortion func ..."
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Abstract—We study the ratedistortion performance of Sparse Regression Codes where the codewords are linear combinations of subsets of columns of a design matrix. It is shown that with minimumdistance encoding and squared error distortion, these codes achieve R∗(D), the Shannon ratedistortion function for i.i.d. Gaussian sources. This completes a previous result which showed that R∗(D) was achievable for distortions below a certain threshold. The proof is based on the second moment method, a popular technique to show that a nonnegative random variable X is strictly positive with high probability. We first identify the reason behind the failure of the vanilla second moment method for this problem, and then introduce a refinement to show that R∗(D) is achievable for all distortions. I.