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53
ArcConsistency and ArcConsistency Again
 Artificial Intelligence
, 1994
"... Constraint networks are known as a useful way to formulate problems such as design, scene labeling, temporal reasoning, and more recently natural language parsing. The problem of the existence of solutions in a constraint network is NPcomplete. Hence, consistency techniques have been widely studied ..."
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Cited by 151 (12 self)
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Constraint networks are known as a useful way to formulate problems such as design, scene labeling, temporal reasoning, and more recently natural language parsing. The problem of the existence of solutions in a constraint network is NPcomplete. Hence, consistency techniques have been widely studied to simplify constraint networks before or during the search of solutions. Arcconsistency is the most used of them. Mohr and Henderson [Moh&Hen86] have proposed AC4, an algorithm having an optimal worstcase time complexity. But it has two drawbacks: its space complexity and its average time complexity. In problems with many solutions, where the size of the constraints is large, these drawbacks become so important that users often replace AC4 by AC3 [Mac&Fre85], a nonoptimal algorithm. In this paper, we propose a new algorithm, AC6, which keeps the optimal worstcase time complexity of AC4 while working out the drawback of space complexity. More, the average time complexity of AC6 is optimal for constraint networks where nothing is known about the semantic of the constraints. At the end of the paper, experimental results show how much AC6 outperforms AC3 and AC4. 1.
An Optimal Coarsegrained Arc Consistency Algorithm
 Artificial Intelligence
"... The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effective fashion. There are two classes of propagation algorithms for general constraints: finegrained algorithms where the removal of a val ..."
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Cited by 93 (16 self)
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The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effective fashion. There are two classes of propagation algorithms for general constraints: finegrained algorithms where the removal of a value for a variable will be propagated to the corresponding values for other variables, and coarsegrained algorithms where the removal of a value will be propagated to the related variables. One big advantage of coarsegrained algorithms, like AC3, over finegrained algorithms, like AC4, is the ease of integration when implementing an algorithm in a constraint solver. However, finegrained algorithms usually have optimal worst case time complexity while coarsegrained algorithms don’t. For example, AC3 is an algorithm with nonoptimal worst case complexity although it is simple, efficient in practice, and widely used. In this paper we propose a coarsegrained algorithm, AC2001/3.1, that is worst case optimal and preserves as much as possible the ease of its integration into a solver (no heavy data structure to be maintained during search). Experimental results show that AC2001/3.1 is competitive with the best finegrained algorithms such as AC6. The idea behind the new algorithm can immediately be applied to obtain a path consistency algorithm that has the bestknown time and space complexity. The same idea is then extended to nonbinary constraints. Preliminary versions of this paper appeared in [BR01, ZY01].
Constraint propagation
 Handbook of Constraint Programming
, 2006
"... Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent ..."
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Cited by 77 (5 self)
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Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent
Using Constraint Metaknowledge to Reduce Arc Consistency Computation
 Artificial Intelligence
, 1999
"... Constraint satisfaction problems are widely used in articial intelligence. They involve nding values for problem variables subject to constraints that specify which combinations of values are consistent. Knowledge about properties of the constraints can permit inferences that reduce the cost of cons ..."
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Cited by 62 (8 self)
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Constraint satisfaction problems are widely used in articial intelligence. They involve nding values for problem variables subject to constraints that specify which combinations of values are consistent. Knowledge about properties of the constraints can permit inferences that reduce the cost of consistency checking. In particular, such inferences can be used to reduce the number of constraint checks required in establishing arc consistency, a fundamental constraintbased reasoning technique. A general ACInference algorithm schema is presented and various forms of inference discussed. A specific algorithm, AC7, is presented, which takes advantage of a simple property common to all binary constraints to eliminate constraint checks that other arc consistency algorithms perform. The effectiveness of this approach is demonstrated analytically, and experimentally.
The design and experimental analysis of algorithms for temporal reasoning
 Journal of Artificial Intelligence Research
, 1996
"... Many applicationsfrom planning and scheduling to problems in molecular biology rely heavily on a temporal reasoning component. In this paper, we discuss the design and empirical analysis of algorithms for a temporal reasoning system based on Allen's in uential intervalbased framework for rep ..."
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Cited by 57 (0 self)
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Many applicationsfrom planning and scheduling to problems in molecular biology rely heavily on a temporal reasoning component. In this paper, we discuss the design and empirical analysis of algorithms for a temporal reasoning system based on Allen's in uential intervalbased framework for representing temporal information. At the core of the system are algorithms for determining whether the temporal information is consistent, and, if so, nding one or more scenarios that are consistent with the temporal information. Two important algorithms for these tasks are a path consistency algorithm and a backtracking algorithm. For the path consistency algorithm, we develop techniques that can result in up to a tenfold speedup over an already highly optimized implementation. For the backtracking algorithm, we develop variable and value ordering heuristics that are shown empirically to dramatically improve the performance of the algorithm. As well, we show that a previously suggested reformulation of the backtracking search problem can reduce the time and space requirements of the backtracking search. Taken together, the techniques we develop allow a temporal reasoning component tosolve problems that are of practical size. 1.
The Constrainedness of Arc Consistency
 in Proceedings of CP97
, 1997
"... . We show that the same methodology used to study phase transition behaviour in NPcomplete problems works with a polynomial problem class: establishing arc consistency. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NPcomplete proble ..."
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Cited by 50 (10 self)
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. We show that the same methodology used to study phase transition behaviour in NPcomplete problems works with a polynomial problem class: establishing arc consistency. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NPcomplete problems, predicts the location of a phase transition in establishing arc consistency. A complexity peak for the AC3 algorithm is associated with this transition. Finite size scaling models both the scaling of this transition and the computational cost. On problems at the phase transition, this model of computational cost agrees with the theoretical worst case. As with NPcomplete problems, constrainedness  and proxies for it which are cheaper to compute  can be used as a heuristic for reducing the number of checks needed to establish arc consistency in AC3. 1 Introduction Following [4] there has been considerable research into phase transition behaviour in NPcomplete problems. Problems from...
Why AC3 is almost always better than AC4 for establishing arc consistency in CSPs
, 1993
"... On the basis of its optimal asymptotic time complexity, AC4 is often considered the best algorithm for establishing arc consistency in constraint satisfaction problems (CSPs). In the present work, AC3 was found to be much more efficient than AC4, for CSPs with a variety of features. (Variable pai ..."
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Cited by 42 (0 self)
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On the basis of its optimal asymptotic time complexity, AC4 is often considered the best algorithm for establishing arc consistency in constraint satisfaction problems (CSPs). In the present work, AC3 was found to be much more efficient than AC4, for CSPs with a variety of features. (Variable pairs were in lexical order, and in AC3 they were added to the end of the list of pairs.) This is supported by arguments for the superiority of AC3 over most of the range of constraint satisfiabilities and for the unlikelihood of conditions leading to worstcase performance. The efficiency of AC4 is affected by the order of variable testing in Phase 1 ('setting up ' phase); performance in this phase can thus be enhanced, and this establishes initial conditions for Phase 2 that improve its performance. But, since AC3 is improved by the same orderings, it still outperforms AC4 in most cases. 1
Speeding up constraint propagation
 In Wallace [14
, 2004
"... Abstract. This paper presents a model and implementation techniques for speeding up constraint propagation. Two fundamental approaches to improving constraint propagation are explored: keeping track of which propagators are at fixpoint, and choosing which propagator to apply next. We show how idempo ..."
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Cited by 27 (5 self)
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Abstract. This paper presents a model and implementation techniques for speeding up constraint propagation. Two fundamental approaches to improving constraint propagation are explored: keeping track of which propagators are at fixpoint, and choosing which propagator to apply next. We show how idempotence reasoning and events help track fixpoints more accurately. We improve these methods by using them dynamically (taking into account current domains to improve accuracy). We define prioritybased approaches to choosing a next propagator and show that dynamic priorities can improve propagation. We illustrate that the use of multiple propagators for the same constraint can be advantageous with priorities, and introduce staged propagators which combine the effects of multiple propagators with priorities for greater efficiency. 1
Exploiting multidirectionality in coarsegrained arc consistency algorithms
 In Proc. of CP’03
, 2003
"... Abstract. Arc consistency plays a central role in solving Constraint Satisfaction Problems. This is the reason why many algorithms have been proposed to establish it. Recently, an algorithm called AC2001 and AC3.1 has been independently presented by their authors. This algorithm which is considered ..."
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Cited by 23 (12 self)
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Abstract. Arc consistency plays a central role in solving Constraint Satisfaction Problems. This is the reason why many algorithms have been proposed to establish it. Recently, an algorithm called AC2001 and AC3.1 has been independently presented by their authors. This algorithm which is considered as a refinement of the basic algorithm AC3 has the advantage of being simple and competitive. However, it does not take into account constraint bidirectionality as AC7 does. In this paper, we address this issue, and, in particular, introduce two new algorithms called AC3.2 and AC3.3 which benefit from good properties of both AC3 and AC7. Indeed, AC3.2 and AC3.3 are as easy to implement as AC3 and take advantage of bidirectionality as AC7 does. More precisely, AC3.2 is a general algorithm which partially exploits bidirectionality whereas AC3.3 is a binary algorithm which fully exploits bidirectionality. It turns out that, when Maintaining Arc Consistency during search, MAC3.2, due to a memorization effect, is more efficient than MAC3.3 both in terms of constraint checks and cpu time. Compared to MAC2001/3.1, our experimental results show that MAC3.2 saves about 50% of constraint checks and, on average, 15 % of cpu time. 1
AC3_d an Efficient ArcConsistency Algorithm with a Low SpaceComplexity
, 2002
"... Arcconsistency algorithms are widely used to prune the searchspace of Constraint Satisfaction Problems (CSPs). They use supportchecks to find out about the properties of CSPs. They use archeuristics to select the constraint and domainheuristics to select the values for their next supportcheck. ..."
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Cited by 17 (7 self)
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Arcconsistency algorithms are widely used to prune the searchspace of Constraint Satisfaction Problems (CSPs). They use supportchecks to find out about the properties of CSPs. They use archeuristics to select the constraint and domainheuristics to select the values for their next supportcheck. We will demonstrate that domainheuristics can significantly enhance the average timecomplexity of existing arcconsistency algorithms. We will combine Alan Mackworth's AC3 and John Gaschnig's DEE and equip the resulting hybrid with a doublesupport domainheuristic thereby creating an arcconsistency algorithm called AC3_d , which has an average timecomplexity which can compete with AC7 and which improves on AC7's spacecomplexity. AC3_d is easy to implement and requires the same data structures as AC3. We will present experimental results to justify our average timecomplexity claim.