Results 1  10
of
35
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 ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
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,
Lossy source encoding via messagepassing and decimation over generalized codewords of LDGM codes
 In: Proceedings of the International Symposium on Information Theory
, 2005
"... Abstract — We describe messagepassing and decimation approaches for lossy source coding using lowdensity generator matrix (LDGM) codes. In particular, this paper addresses the problem of encoding a Bernoulli() source: for randomly generated LDGM codes with suitably irregular degree distributions, ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
Abstract — We describe messagepassing and decimation approaches for lossy source coding using lowdensity generator matrix (LDGM) codes. In particular, this paper addresses the problem of encoding a Bernoulli() source: for randomly generated LDGM codes with suitably irregular degree distributions, our methods yield performance very close to the rate distortion limit over a range of rates. Our approach is inspired by the survey propagation (SP) algorithm, originally developed by Mézard et al. [1] for solving random satisfiability problems. Previous work by Maneva et al. [2] shows how SP can be understood as belief propagation (BP) for an alternative representation of satisfiability problems. In analogy to this connection, our approach is to define a family of Markov random fields over generalized codewords, from which local messagepassing rules can be derived in the standard way. The overall source encoding method is based on messagepassing, setting a subset of bits to their preferred values (decimation), and reducing the code. I.
The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies
, 2006
"... 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 ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
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
Leveraging belief propagation, backtrack search, and statistics for model counting
"... Abstract. We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different tradeoffs between efficiency, bound quality, and correctness guarantee ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
Abstract. We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different tradeoffs between efficiency, bound quality, and correctness guarantee. For lower bounds, we use a recent framework for probabilistic correctness guarantees, and exploit message passing techniques for marginal probability estimation, namely, variations of Belief Propagation (BP). Our results suggest that BP provides useful information even on structured loopy formulas. For upper bounds, we perform multiple runs of the MiniSat SAT solver with a minor modification, and obtain statistical bounds on the model count based on the observation that the distribution of a certain quantity of interest is often very close to the normal distribution. Our experiments demonstrate that our model counters based on these two ideas, BPCount and MiniCount, can provide very good bounds in time significantly less than alternative approaches. 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 ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
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
Survey Propagation Revisited
"... Survey propagation (SP) is an exciting new technique that has been remarkably successful at solving very large hard combinatorial problems, such as determining the satisfiability of Boolean formulas. In a promising attempt at understanding the success of SP, it was recently shown that SP can be view ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Survey propagation (SP) is an exciting new technique that has been remarkably successful at solving very large hard combinatorial problems, such as determining the satisfiability of Boolean formulas. In a promising attempt at understanding the success of SP, it was recently shown that SP can be viewed as a form of belief propagation, computing marginal probabilities over certain objects called covers of a formula. This explanation was, however, shortly dismissed by experiments suggesting that nontrivial covers simply do not exist for large formulas. In this paper, we show that these experiments were misleading: not only do covers exist for large hard random formulas, SP is surprisingly accurate at computing marginals over these covers despite the existence of many cycles in the formulas. This reopens a potentially simpler line of reasoning for understanding SP, in contrast to some alternative lines of explanation that have been proposed assuming covers do not exist. 1
Modern coding theory: the statistical mechanics and computer science point of view
, 2007
"... These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress c ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as ‘large graphical models’. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field.
Computing pure Nash equilibria via Markov random fields
 In ACMEC
, 2006
"... In this paper we present a novel generic mapping between Graphical Games and Markov Random Fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Thus, the problem of deciding whether a graphical game has a pure Nash equilibrium, a wellknown intractab ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
In this paper we present a novel generic mapping between Graphical Games and Markov Random Fields so that pure Nash equilibria in the former can be found by statistical inference on the latter. Thus, the problem of deciding whether a graphical game has a pure Nash equilibrium, a wellknown intractable problem, can be attacked by wellestablished algorithms such as Belief Propagation, Junction Trees, Markov Chain Monte Carlo and Simulated Annealing. Large classes of graphical games become thus tractable, including all classes already known, but also new classes such as the games with O(log n) treewidth.
MessagePassing and Local Heuristics as Decimation Strategies for Satisfiability
"... Decimation is a simple process for solving constraint satisfaction problems, by repeatedly fixing variable values and simplifying without reconsidering earlier decisions. We investigate different decimation strategies, contrasting those based on local, syntactic information from those based on messa ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Decimation is a simple process for solving constraint satisfaction problems, by repeatedly fixing variable values and simplifying without reconsidering earlier decisions. We investigate different decimation strategies, contrasting those based on local, syntactic information from those based on message passing, such as statistical physics based Survey Propagation (SP) and the related and more wellknown Belief Propagation (BP). Our results reveal that once we resolve convergence issues, BP itself can solve fairly hard random kSAT formulas through decimation; the gap between BP and SP narrows down quickly as k increases. We also investigate observable differences between BP/SP and other common CSP heuristics as decimation proceeds, exploring the hardness of the decimated formulas and identifying a somewhat unexpected feature of message passing heuristics, namely, unlike other heuristics for satisfiability, they avoid unit propagation as variables are fixed.
Incomplete Algorithms
, 2008
"... An incomplete method for solving the propositional satisfiability problem (or a general constraint satisfaction problem) is one that does not provide the guarantee that it will eventually either report a satisfying assignment or declare that the given formula is unsatisfiable. In practice, most such ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
An incomplete method for solving the propositional satisfiability problem (or a general constraint satisfaction problem) is one that does not provide the guarantee that it will eventually either report a satisfying assignment or declare that the given formula is unsatisfiable. In practice, most such methods are biased towards the satisfiable side: they are typically run with a preset resource limit, after which they either produce a valid solution or report failure; they never declare the formula to be unsatisfiable. These are the kind of algorithms we will discuss in this chapter. In complexity theory terms, such algorithms are referred to as having onesided error. In principle, an incomplete algorithm could instead be biased towards the unsatisfiable side, always providing proofs of unsatisfiability but failing to find solutions to some satisfiable instances, or be incomplete with respect to both satisfiable and unsatisfiable instances (and thus have twosided error). Unlike systematic solvers often based on an exhaustive branching and backtracking search, incomplete methods are generally based on stochastic local search,