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23
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 constrain ..."
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Cited by 116 (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 ...
Satbased analysis of feature models is easy
 In 13th International Conference on Software Product Lines (SPLC 2009
, 2009
"... Feature models are a popular variability modeling notation used in product line engineering. Automated analyses of feature models, such as consistency checking and interactive or offline product selection, often rely on translating models to propositional logic and using satisfiability (SAT) solvers ..."
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Cited by 40 (6 self)
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Feature models are a popular variability modeling notation used in product line engineering. Automated analyses of feature models, such as consistency checking and interactive or offline product selection, often rely on translating models to propositional logic and using satisfiability (SAT) solvers. Efficiency of individual satisfiabilitybased analyses has been reported previously. We generalize and quantify these studies with a series of independent experiments. We show that previously reported efficiency is not incidental. Unlike with the general SAT instances, which fall into easy and hard classes, the instances induced by feature modeling are easy throughout the spectrum of realistic models. In particular, the phenomenon of phase transition is not observed for realistic feature models. Our main practical conclusion is a general encouragement for researchers to continued development of SATbased methods to further exploit this efficiency in future. 1
Frozen Development in Graph Coloring
 THEORETICAL COMPUTER SCIENCE
, 2000
"... We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the a ..."
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Cited by 34 (5 self)
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We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10^300. We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...
Exact Phase Transitions in Random Constraint Satisfaction Problems
 Journal of Artificial Intelligence Research
, 2000
"... In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems are satis able to a region where almost all problems are unsatis able do exist for Model RB as the number ..."
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Cited by 32 (9 self)
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In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems are satis able to a region where almost all problems are unsatis able do exist for Model RB as the number of variables approaches in nity. Moreover, the critical values at which the phase transitions occur are also known exactly. By relating the hardness of Model RB to Model B, it is shown that there exist a lot of hard instances in Model RB.
Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances
 in Theoretical Computer Science
, 2003
"... This paper analyzes the resolution complexity of two random CSP models, i.e. Model RB/RD for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, this paper proves that almost all instances of Model RB/RD have no tre ..."
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Cited by 18 (5 self)
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This paper analyzes the resolution complexity of two random CSP models, i.e. Model RB/RD for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, this paper proves that almost all instances of Model RB/RD have no treelike resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of ProofComplexity theory, but also propose models with both many hard instances and exact phase transitions. Moreover, it is shown both theoretically and experimentally that an application of RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating oneway functions. Finally, conclusions are presented, as well as a detailed comparison of RB/RD with some wellstudied models such as the Hamiltonian cycle problem and random 3SAT.
Well out of reach: Why hard problems are hard
 APES RESEARCH GROUP
, 1999
"... We show that problems at the uncolorability phase transition are well out of reach of intelligent algorithms. Since there are not small and easily checkable subgraphs which can be used to confirm uncolorability quickly, we cannot hope to build more intelligent algorithms to avoid hard problems at t ..."
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Cited by 16 (5 self)
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We show that problems at the uncolorability phase transition are well out of reach of intelligent algorithms. Since there are not small and easily checkable subgraphs which can be used to confirm uncolorability quickly, we cannot hope to build more intelligent algorithms to avoid hard problems at the phase transition. Also, our results suggest that a conjectured double phase transition in graph coloring occurs only in small graphs. Similar results are likely in other NPcomplete problems where instances from phase transitions are hard for all known algorithms, and will help to explain the phenomenon. Furthermore, our results help to elucidate the distinction between polynomial and nonpolynomial search behavior.
An Analysis of Phase Transition in NK Landscapes
 Journal of Artificial Intelligence Research
, 2002
"... In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is ..."
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Cited by 10 (2 self)
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In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
 Journal of Artificial Intelligence Research
, 2004
"... In recent years, there has been much interest in phase transitions of combinatorial problems. ..."
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Cited by 9 (2 self)
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In recent years, there has been much interest in phase transitions of combinatorial problems.