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24
The Constrainedness of Search
 In Proceedings of AAAI96
, 1999
"... We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition ..."
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Cited by 117 (26 self)
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We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition of constrainedness generalizes a number of parameters used to study phase transition behaviour in a wide variety of problem domains. As well as predicting the location of phase transitions in solubility, constrainedness provides insight into why problems at phase transitions tend to be hard to solve. Such problems are on a constrainedness "knifeedge", and we must search deep into the problem before they look more or less soluble. Heuristics that try to get off this knifeedge as quickly as possible by, for example, minimizing the constrainedness are often very effective. We show that heuristics from a wide variety of problem domains can be seen as minimizing the constrainedness (or proxies ...
Phase Transitions and Annealed Theories: Number Partitioning as a Case Study
 In Proceedings of ECAI96
, 1996
"... . We outline a technique for studying phase transition behaviour in computational problems using number partitioning as a case study. We first build an "annealed" theory that assumes independence between parts of the number partition problem. Using this theory, we identify a parameter which represen ..."
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Cited by 32 (10 self)
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. We outline a technique for studying phase transition behaviour in computational problems using number partitioning as a case study. We first build an "annealed" theory that assumes independence between parts of the number partition problem. Using this theory, we identify a parameter which represents the "constrainedness" of a problem. We determine experimentally the critical value of this parameter at which a rapid transition between soluble and insoluble problems occurs. Finitesize scaling methods developed in statistical mechanics describe the behaviour around the critical value. We identify phase transition behaviour in both the decision and optimization versions of number partitioning, in the size of the optimal partition, and in the quality of heuristic solutions. This case study demonstrates how annealed theories and finitesize scaling allows us to compare algorithms and heuristics in a precise and quantitative manner. 1 Introduction Phase transition behaviour has recently r...
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
Analysis of heuristics for number partitioning
 Computational Intelligence
, 1998
"... We illustrate the use of phase transition behavior in the study of heuristics. Using an “annealed ” theory, we define a parameter that measures the “constrainedness ” of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show t ..."
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Cited by 25 (11 self)
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We illustrate the use of phase transition behavior in the study of heuristics. Using an “annealed ” theory, we define a parameter that measures the “constrainedness ” of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show that constrainedness can be used to analyze and compare algorithms and heuristics for number partitioning in a precise and quantitative manner. For example, we demonstrate that on uniform random problems both the Karmarkar–Karp and greedy heuristics minimize the constrainedness, but that the decisions made by the Karmarkar–Karp heuristic are superior at reducing constrainedness. This supports the better performance observed experimentally for the Karmarkar–Karp heuristic. Our results refute a conjecture of Fu that phase transition behavior does not occur in number partitioning. Additionally, they demonstrate that phase transition behavior is useful for more than just simple benchmarking. It can, for instance, be used to analyze heuristics, and to compare the quality of heuristic solutions. Key words: heuristics, number partitioning, phase transitions. 1.
The Phase Transition Behaviour of Maintaining Arc Consistency
 In Proceedings of ECAI96
, 1995
"... In this paper, we study two recently presented algorithms employing a "full lookahead" strategy: MAC (Maintaining Arc Consistency); and the hybrid MACCBJ, which combines conflictdirected backjumping capability with MAC. We observe their behaviour with respect to the phase transition properties of ..."
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Cited by 24 (6 self)
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In this paper, we study two recently presented algorithms employing a "full lookahead" strategy: MAC (Maintaining Arc Consistency); and the hybrid MACCBJ, which combines conflictdirected backjumping capability with MAC. We observe their behaviour with respect to the phase transition properties of randomlygenerated binary constraint satisfaction problems, and investigate the benefits of maintaining a higher level of consistency during search by comparing MAC and MACCBJ with the FC and FCCBJ algorithms, which maintain only node consistency. The phase transition behaviour that has been observed for many classes of problem as a control parameter is varied has prompted a flurry of research activity in recent years. Studies of these transitions, from regions where most problems are easy and soluble to regions where most are easy but insoluble, have raised a number of important issues such as the phenomenon of exceptionally hard problems ("ehps") in the easysoluble region, and the grow...
A Duality between Clause Width and Clause Density for SAT
 In IEEE Conference on Computational Complexity (CCC
"... We consider the relationship between the complexities of and those of restricted to formulas of constant density. Let be the infimum of those such that on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for a ..."
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Cited by 22 (3 self)
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We consider the relationship between the complexities of and those of restricted to formulas of constant density. Let be the infimum of those such that on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for any, can be solved in time independent of if and only if the same is true for with any fixed density of clauses to variables. We derive some interesting consequences from this. For example, assuming thatis exponentially hard (that is,), of any fixed density can be solved in time whose exponent is strictly less than that for general. We also give an improvement to the sparsification lemma of [12] showing that instances of of density slightly more than exponential in are almost the hardest instances of. The previous result showed this for densities doubly exponential in. 1.
Coreduction homology algorithm
 Discrete & Computational Geometry
"... Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, ..."
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Cited by 18 (8 self)
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Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates. 1.
The search for Satisfaction
, 1999
"... In recent years, there has been an explosion of research in AI into propositional satis ability (or Sat). There are many factors behind the increased interest in this area. One factor is the improvement in search procedures for Sat. New local search procedures like Gsat are able to solve Sat problem ..."
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Cited by 14 (1 self)
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In recent years, there has been an explosion of research in AI into propositional satis ability (or Sat). There are many factors behind the increased interest in this area. One factor is the improvement in search procedures for Sat. New local search procedures like Gsat are able to solve Sat problems with thousands of variables. At the same time, implementations of complete search algorithms like DavisPutnam have been able to solve open mathematical problems. Another factor is the identi cation of hard Sat problems at a phase transition in solubility. A third factor is the demonstration that we can often solve real world problems by encoding them into Sat. There has also seen an improved theoretical understanding of Sat, particularly in the analysis of such phase transition behaviour. This paper reviews the state of the art for research into satis ability, and discuss applications in which algorithms for satis ability have proved successful.