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47
A New Look at Survey Propagation and its Generalizations
"... We study the survey propagation algorithm [19, 5, 4], which is an iterative technique that appears to be very effective in solving random kSAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), ..."
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Cited by 49 (13 self)
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We study the survey propagation algorithm [19, 5, 4], which is an iterative technique that appears to be very effective in solving random kSAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number ρ. We then show that applying belief propagation— a wellknown “messagepassing” technique—to this family of MRFs recovers various algorithms, ranging from pure survey propagation at one extreme (ρ = 1) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme (ρ = 0). Configurations in these MRFs have a natural interpretation as generalized satisfiability assignments, on which a partial order can be defined. We isolate cores as minimal elements in this partial
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 22 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
Reconstruction for models on random graphs
 In FOCS ’07: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
, 2007
"... The reconstruction problem requires to estimate a random variable given ‘far away ’ observations. Several theoretical results (and simple algorithms) are available when the underlying probability distribution is Markov with respect to a tree. In this paper we estabilish several exact thresholds for ..."
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Cited by 17 (4 self)
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The reconstruction problem requires to estimate a random variable given ‘far away ’ observations. Several theoretical results (and simple algorithms) are available when the underlying probability distribution is Markov with respect to a tree. In this paper we estabilish several exact thresholds for loopy graphs. More precisely we consider models on random graphs that converge locally to trees. We establish the reconstruction thresholds for the Ising model both with attractive and random interactions (respectively, ‘ferromagnetic ’ and ‘spin glass’). Remarkably, in the first case the result does not coincide with the corresponding tree threshold. Among the other tools, we develop a sufficient condition for the tree and graph reconstruction problem to coincide. We apply such condition to antiferromagnetic colorings of random graphs. 1 Introduction and
The connectivity of boolean satisfiability: Computational and structural dichotomies
 in Proc. 33 rd Intl. Colloquium on Automata, Languages and Programming
"... Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we ..."
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Cited by 15 (4 self)
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Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivityrelated properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer’s framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and stconnectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACEcomplete, while the tractable side which includes but is not limited to all problems with polynomial time algorithms for satisfiability is in P for the stconnectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACEcomplete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary. 1
BeliefPropagation for Weighted bMatchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
 in arXiv, http://www.arxiv.org/abs/0709.1190v1
, 2007
"... We consider the general problem of finding the minimum weight bmatching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. This result is notabl ..."
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Cited by 13 (0 self)
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We consider the general problem of finding the minimum weight bmatching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. This result is notable in several regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure. (2) Instead of showing that BP leads to a PTAS, we give a finite bound for the number of iterations after which BP has converged to the exact solution. (3) Variants of the proof work for both synchronous and asynchronous BP; to the best of our knowledge, it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem. (4) It works for both ordinary bmatchings and the more difficult case of perfect bmatchings. (5) Together with the recent work of Sanghavi, Malioutov and Wilskly [41] they are the first complete proofs showing that tightness of LP implies correctness of BP. 1
Finite size scaling for the core of large random hypergraphs
, 2008
"... The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of lowdensity paritycheck codes used over the binary erasure channel. Si ..."
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Cited by 7 (4 self)
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The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of lowdensity paritycheck codes used over the binary erasure channel. Similar structures emerge in a variety of NPhard combinatorial optimization and decision problems, from vertex cover to satisfiability. For a uniformly chosen random hypergraph of m = nρ vertices and n hyperedges, each consisting of the same fixed number l ≥ 3 of vertices, the size of the core exhibits for large n a firstorder phase transition, changing from o(n) for ρ>ρc to a positive fraction of n for ρ<ρc, with a transition window size �(n −1/2) around ρc> 0. Analyzing the corresponding “leaf removal” algorithm, we determine the associated finitesize scaling behavior. In particular, if ρ is inside the scaling window (more precisely, ρ = ρc + rn −1/2), the probability of having a core of size �(n) has a limit strictly between 0 and 1, and a leading correction of order �(n −1/6). The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with n. Thisbehavioris expected to be universal for a wide collection of combinatorial problems.
Gibbs Measures and Phase Transitions on Sparse Random Graphs
"... Abstract: Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these dist ..."
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Cited by 6 (3 self)
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Abstract: Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.
Reconstruction and Clustering in Random Constraint Satisfaction Problems
, 2009
"... Random instances of Constraint Satisfaction Problems (CSP’s) appear to be hard for all known algorithms, when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we for ..."
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Cited by 5 (3 self)
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Random instances of Constraint Satisfaction Problems (CSP’s) appear to be hard for all known algorithms, when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSP’s, and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSP’s. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as, random instances of NotAllEqualSAT, kXOR formulae, hypergraph 2coloring, and graph kcoloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of problems with a similar threshold structure.