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58
Typical random 3SAT formulae and the satisfiability threshold
 in Proceedings of the Eleventh ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to o ..."
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Cited by 87 (2 self)
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Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to other problems, such as the 3colourability of random graphs. 1
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 46 (12 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
On the solutionspace geometry of random constraint satisfaction problems
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... For a large number of random constraint satisfaction problems, such as random kSAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomialtime algorithms for these problems fail to find solutions ..."
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Cited by 45 (2 self)
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For a large number of random constraint satisfaction problems, such as random kSAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomialtime algorithms for these problems fail to find solutions even at much lower densities. To understand the origin of this gap we study how the structure of the space of solutions evolves in such problems as constraints are added. In particular, we prove that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. Moreover, inside each cluster most variables are frozen, i.e., take only one value. The existence of such frozen variables gives a satisfying intuitive explanation for the failure of the polynomialtime algorithms analyzed so far. At the same time, our results establish rigorously one of the two main hypotheses underlying Survey Propagation, a heuristic introduced by physicists in recent years that appears to perform extraordinarily well on random constraint satisfaction problems. 1
A Spectral Technique for Random Satisfiable 3CNF Formulas
, 2002
"... Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x_1, ..., x_n uniformly at random and including every clause with i literals set true by φ with probability p_i, independently. We show that for any 0 ≤ η_2, η_3 ≤ 1 there is a constant d_mi ..."
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Cited by 31 (3 self)
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Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x_1, ..., x_n uniformly at random and including every clause with i literals set true by φ with probability p_i, independently. We show that for any 0 ≤ η_2, η_3 ≤ 1 there is a constant d_min so that for all d ≥ d_min a spectral algorithm similar to the graph coloring algorithm of [1] will find a satisfying assignment with high probability for p_1 = d/n², p_2 = ...
The satisfiability threshold for random 3SAT is at least 3.52
, 2003
"... We prove that a random 3SAT instance with clausetovariable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement. 1 ..."
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Cited by 31 (1 self)
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We prove that a random 3SAT instance with clausetovariable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement. 1
Survey propagation: an algorithm for satisfiability
, 2002
"... ABSTRACT: We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M/N is close to the experimental threshold αc separating the region where almost all form ..."
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Cited by 30 (1 self)
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ABSTRACT: We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M/N is close to the experimental threshold αc separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when α is close to (but smaller than) αc. We introduce a new type of message passing algorithm which allows to find efficiently a satisfying assignment of the variables in this difficult region. This algorithm is iterative and composed of two main parts. The first is a messagepassing procedure which generalizes the usual methods like SumProduct or Belief Propagation: It passes messages that may be thought of as surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional
Heuristics for fast exact model counting
 In Proc. 8th International Conference on Theory and Applications of Satisfiability Testing
, 2005
"... Abstract. An important extension of satisfiability testing is modelcounting, a task that corresponds to problems such as probabilistic reasoning and computing the permanent of a Boolean matrix. We recently introduced Cachet, an exact modelcounting algorithm that combines formula caching, clause le ..."
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Cited by 29 (2 self)
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Abstract. An important extension of satisfiability testing is modelcounting, a task that corresponds to problems such as probabilistic reasoning and computing the permanent of a Boolean matrix. We recently introduced Cachet, an exact modelcounting algorithm that combines formula caching, clause learning, and component analysis. This paper reports on experiments with various techniques for improving the performance of Cachet, including componentselection strategies, variableselection branching heuristics, randomization, backtracking schemes, and crosscomponent implications. The result of this work is a highlytuned version of Cachet, the first (and currently, only) system able to exactly determine the marginal probabilities of variables in random 3SAT formulas with 150+ variables. We use this to discover an interesting property of random formulas that does not seem to have been previously observed. 1
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 24 (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,
Selecting Complementary Pairs of Literals
, 2003
"... We analyze an algorithm that in each free step Selects & Sets to a value a pair of complementary literals of specified corresponding degrees. Unit Clauses, if they exist, are given priority. This algorithm is proved to succeed on input random 3CNF formulas of density c3 = 3.52+, establishing th ..."
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Cited by 21 (2 self)
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We analyze an algorithm that in each free step Selects & Sets to a value a pair of complementary literals of specified corresponding degrees. Unit Clauses, if they exist, are given priority. This algorithm is proved to succeed on input random 3CNF formulas of density c3 = 3.52+, establishing that the conjectured threshold value for random 3CNF formulas is at least 3.52+.
Linear upper bounds for random walk on small density random 3cnfs
 IN PROC. 44TH IEEE SYMP. ON FOUND. OF COMP. SCIENCE
, 2003
"... We analyze the efficiency of the random walk algorithmon random 3CNF instances, and prove linear upper boundson the running time of this algorithm for small clause density, less than 1:63. Our upper bound matches the observedrunning time to within a multiplicative factor. This is the first subex ..."
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Cited by 20 (1 self)
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We analyze the efficiency of the random walk algorithmon random 3CNF instances, and prove linear upper boundson the running time of this algorithm for small clause density, less than 1:63. Our upper bound matches the observedrunning time to within a multiplicative factor. This is the first subexponential upper bound on the running time of alocal improvement algorithm on random instances. Our proof introduces a simple, yet powerful tool for analyzing such algorithms, which may be of further use. This object, called a terminator, is a weighted satisfying assignment. We show that any CNF having a good (small weight) terminator, is assured to be solved quickly by the randomwalk algorithm. This raises the natural question of the terminator threshold which is the maximal clause density forwhich such assignments exist (with high probability). We use the analysis of the pure literal heuristic presentedby Broder, Frieze and Upfal [12, 22] and show that for small clause densities good terminators exist. Thus we showthat the Pure Literal threshold ( ss 1:63) is a lower boundon the terminator threshold. (We conjecture the terminator threshold to be in fact higher). One nice property of terminators is that they can befound efficiently, via linear programming. This makes