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15
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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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
Enforcing Arc Consistency using Bitwise Operations
 CONSTRAINT PROGRAMMING LETTERS
, 2007
"... In this paper, we propose to exploit bitwise operations to speed up some important computations such as looking for a support of a value in a constraint, or determining if a value is substitutable by another one. Considering a computer equipped with a xbit CPU, one can then expect an increase of th ..."
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Cited by 9 (5 self)
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In this paper, we propose to exploit bitwise operations to speed up some important computations such as looking for a support of a value in a constraint, or determining if a value is substitutable by another one. Considering a computer equipped with a xbit CPU, one can then expect an increase of the performance by a coefficient up to x (which may be important, since x is equal to 32 or 64 in many current central units). To show the interest of enforcing arc consistency using bitwise operations, we introduce a new variant of AC3, denoted by AC3 bit, which can be used when constraints are (or can be) represented in extension. This new algorithm when embedded in MAC, is approximately two times more efficient than AC3 rm. Note that AC3 rm is a variant of AC3 which exploits the concept of residual supports and has been shown to be faster than AC2001.
Exploiting constraint weights for revision ordering in Arc Consistency Algorithms
 In Submitted to the ECAI08 Workshop on Modeling and Solving Problems with Constraints
, 2008
"... Abstract. Coarse grained arc consistency algorithms, like AC3, operate by maintaining a list of arcs (or variables) that records the revisions that are still to be performed. It is well known that the performance of such algorithms is affected by the order in which revisions are carried out. As a r ..."
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Cited by 7 (3 self)
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Abstract. Coarse grained arc consistency algorithms, like AC3, operate by maintaining a list of arcs (or variables) that records the revisions that are still to be performed. It is well known that the performance of such algorithms is affected by the order in which revisions are carried out. As a result, several heuristics for ordering the elements of the revision list have been proposed. These heuristics exploit information about the original and the current state of the problem, such as domain sizes, variable degrees, and allowed combinations of values, to reduce the number of constraint checks and list operations aiming at speeding up arc consistency computation. Recently, Boussemart et al. proposed novel variable ordering heuristics that exploit information about failures gathered throughout search and recorded in the form of constraint weights. Such heuristics are now considered as the most efficient general purpose variable ordering heuristic for CSPs. In this paper we show how information about constraint weights can be exploited to efficiently order the revision list when AC is applied during search. We propose a number of simple revision ordering heuristics based on constraint weights for arc, variable, and constraint oriented implementations of coarse grained arc consistency algorithms, and compare them to the most efficient existing revision ordering heuristic. Importantly, the new heuristics can not only reduce the numbers of constraints checks and list operations, but also cut down the size of the explored search tree. Results from various structured and random problems demonstrate that some of the proposed heuristics can offer significant speedups. 1
Lightweight ArcConsistency Algorithms
, 2003
"... Arcconsistency algorithms are the workhorse of many backtrack algorithms. Most research on arcconsistency algorithms is focusing on the design of algorithms that are optimal when it comes to worst case scenarios. This report will provide experimental evidence that, despite common belief to the con ..."
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Cited by 6 (2 self)
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Arcconsistency algorithms are the workhorse of many backtrack algorithms. Most research on arcconsistency algorithms is focusing on the design of algorithms that are optimal when it comes to worst case scenarios. This report will provide experimental evidence that, despite common belief to the contrary, the ability to deal efficiently with such worst case scenarios may not be a prerequisite for solving quickly. It will compare on the one hand AC2001 , which has an optimal worst case timecomplexity and is considered efficient, and on the other AC3 d , which is not optimal when it comes to its worst case timecomplexity, but which has a better spacecomplexity than AC2001. Both algorithms will be compared for MAC search and for stand alone arcconsistency (the task of making a single CSP arcconsistent). For stand alone arcconsistency AC3 d is the better algorithm when it comes to time but there is no clear winner when it comes to minimising the number of checks. For search the results are more interesting. MAC2001 is by far the better algorithm when it comes to minimising the number of checks. However, MAC3 d is considerably faster on average. For difficult random problems, that took between minutes and 1.5 hour to solve, MAC3 d was about 1.5 times faster on average than MAC2001. As soon as MAC2001 starts to become successful in avoiding the duplication of many checks it begins to invest much more additional solution time. These observations suggest that being worst case optimal may come at a price of being less efficient on average in search and that algorithms like MAC3 d are promising. Contents 1
Improving ArcConsistency Algorithms with DoubleSupport Checks
 In Proceedings of the Eleventh Irish Conference on Artificial Intelligence and Cognitive Science
, 2000
"... Arcconsistency algorithms are widely used to simplify Constraint Satisfaction Problems. The new notion of a doublesupport check is presented to improve the average performance of arcconsistency algorithms. The improvement is that, where possible, consistencychecks are used to nd supports for t ..."
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Cited by 6 (4 self)
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Arcconsistency algorithms are widely used to simplify Constraint Satisfaction Problems. The new notion of a doublesupport check is presented to improve the average performance of arcconsistency algorithms. The improvement is that, where possible, consistencychecks are used to nd supports for two values, one value in the domain of each variable, which were previously known to be unsupported. It is motivated by the insight that in order to minimize the number of consistencychecks it is necessary to maximize the number of uncertainties which are resolved per check. The idea is used to improve AC3 and DEE and results in a new general purpose arcconsistency algorithm called AC3 b . Experimental results of a comparison of AC3, DEE, AC3 b and AC7 are presented. The results seem to indicate that AC3 b always performs better than DEE and usually performs better than both AC3 and AC7 for the set of testproblems under consideration. 1 Introduction Arcconsistency algorit...
Two new lightweight arc consistency algorithms
 PROCEEDINGS OF THE FIRST INTERNATIONAL WORKSHOP ON CONSTRAINT PROPAGATION AND IMPLEMENTATION (CPAI’2004
, 2004
"... Coarsegrained arc consistency algorithms like AC3, AC3d, and AC2001, are efficient when it comes to transforming a Constraint Satisfaction Problem (CSP) to its arc consistent equivalent. These algorithms repeatedly carry out revisions to remove unsupported values from the domains of the variab ..."
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Cited by 4 (0 self)
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Coarsegrained arc consistency algorithms like AC3, AC3d, and AC2001, are efficient when it comes to transforming a Constraint Satisfaction Problem (CSP) to its arc consistent equivalent. These algorithms repeatedly carry out revisions to remove unsupported values from the domains of the variables. The order of these revisions is determined by socalled revision ordering heuristics. In this paper, we classify revision ordering heuristics into three different categories: arc based, variable based, and reverse variable based revision ordering heuristics. We point out advantages of using reverse variable based revision ordering heuristics and propose two new lightweight arc consistency algorithms AC3dl and AC3ds, which exploit these advantages. Both algorithms are equipped with domain heuristics which are inspired by AC3d’s double support heuristic. AC3dl uses a lazy version of a double support heuristic while AC3ds uses AC3d’s double support heuristic with a minor change. We experimentally compare MAC3, MAC3d, MAC3dl and MAC3ds. MAC3ds is the best in saving checks. MAC3dl is good in saving time and checks on average. Experimental results demonstrate that lightweight algorithms based on reverse variable based revision ordering heuristics are good in saving checks as well as time.
Queue representation for arc consistency algorithms
 PROCEEDINGS OF THE FIFTEENTH IRISH CONFERENCE ON ARTIFICIAL INTELLIGENCE AND COGNITIVE SCIENCE
, 2004
"... Arc consistency algorithms are an indispensable tool for efficiently solving many problems arising in computer science, mathematics, and the real world. Coarsegrained algorithms like AC2001, AC3, and AC3d are efficient when it comes to saving time. Revision ordering heuristics for selecting arc ..."
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Cited by 3 (1 self)
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Arc consistency algorithms are an indispensable tool for efficiently solving many problems arising in computer science, mathematics, and the real world. Coarsegrained algorithms like AC2001, AC3, and AC3d are efficient when it comes to saving time. Revision ordering heuristics for selecting arcs or variables from a data structure called a queue have a major impact on the average performance of these algorithms. This paper demonstrates that queue representation also plays an important role. We study a vqueue (queue containing variables) based on lists and aqueues (queues containing arcs) based on lists, linear probing, and AVL trees. Aqueues based on lists prove the least efficient and vqueues prove the most efficient for saving time. Vqueues result in more checks but less time because of the small overhead of queue maintenance. We experimentally compare MAC3, MAC3d, and MAC2001. For random problems vqueues enable the algorithms to solve about 4 times faster on average than aqueues based on lists. With a vqueue MAC3 becomes the fastest solver on average.
Lightweight MAC Algorithms
, 2003
"... Arcconsistency algorithms are the workhorse of backtrackers that Maintain ArcConsistency (MAC). This report will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arcconsistency algorithm to have an optimal worst case timecomplexity. ..."
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Cited by 3 (1 self)
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Arcconsistency algorithms are the workhorse of backtrackers that Maintain ArcConsistency (MAC). This report will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arcconsistency algorithm to have an optimal worst case timecomplexity. To sacrifice this optimality allows MAC solvers that (1) do not need additional data structures during search, (2) have an excellent average timecomplexity, and (3) have a spacecomplexity which improves significantly on that of MAC solvers that have optimal arcconsistency components. Results will be presented from an experimental comparison between MAC2001, MAC3 d and related algorithms. MAC2001 has an arcconsistency component with an optimal worst case timecomplexity, whereas MAC3 d does not. MAC2001 requires additional data structures during search, whereas MAC3 d does not. MAC3 d has a spacecomplexity of O(e + nd), where n is the number of variables, d the maximum domain size, and e the number of constraints. We shall demonstrate that MAC2001's spacecomplexity is O(ed min(n; d)). MAC2001 required about 35% more solution time on average than MAC3 d for easy and hard random problems. MAC3 d recorded the least solution time for 21 of the 25 realworld problems. Our results indicate that if checks are cheap then lightweight algorithms like MAC3 d are promising.
To avoid repeating checks does not always save time
 In Proceedings of AICS’2003: The 14th Irish Artificial Intelligence and Cognitive Science
, 2003
"... Abstract. Arcconsistency algorithms are the workhorse of backtrackers that Maintain ArcConsistency (MAC). This paper will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arcconsistency algorithm to have an optimal worst case timeco ..."
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Cited by 2 (0 self)
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Abstract. Arcconsistency algorithms are the workhorse of backtrackers that Maintain ArcConsistency (MAC). This paper will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arcconsistency algorithm to have an optimal worst case timecomplexity. To sacrifice this optimality allows MAC solvers that (1) do not need additional data structures during search, (2) have an excellent average timecomplexity, and (3) have a spacecomplexity which improves significantly on that of MAC solvers that have optimal arcconsistency components. Results will be presented from an experimental comparison between MAC2001, MAC3d and related algorithms. MAC2001 has an arcconsistency component with an optimal worst case timecomplexity, whereas MAC3d does not. MAC2001 requires additional data structures during search, whereas MAC3d does not. MAC3d has a spacecomplexity of O(e + nd), where n is the number of variables, d the maximum domain size, and e the number of constraints. We shall demonstrate that MAC2001’s spacecomplexity is O(ed min(n, d)). MAC2001 required about 35 % more solution time on average than MAC3d for easy and hard random problems, MAC3d was faster for 40 % of the realworld problems but slower for the remaining realworld problems. Our results are an indication that if checks are cheap then lightweight algorithms like MAC3d are promising. 1
Nonviability deductions in arcconsistency computation
 in &quot;Proc. of the Nineteenth International Conference on Logic Programming (ICLP 2004)&quot;, Lecture Notes in Computer Science, SpringerVerlag, 2004, p. 343–355. Internal Reports
"... Abstract ArcConsistency (AC) techniques have been used extensively in the study of Constraint Satisfaction Problems (CSP). These techniques are used to simplify the CSP before or during the search for its solutions. Some of the most efficient algorithms for AC computation are AC6++ and AC7. The no ..."
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Abstract ArcConsistency (AC) techniques have been used extensively in the study of Constraint Satisfaction Problems (CSP). These techniques are used to simplify the CSP before or during the search for its solutions. Some of the most efficient algorithms for AC computation are AC6++ and AC7. The novelty of these algorithms is that they satisfy the socalled four desirable properties for AC computation. The main purpose of these interesting properties is to reduce as far as possible the number of constraint checks during AC computation while keeping a reasonable space complexity. In this paper we prove that, despite providing a remarkable reduction in the number of constraint checks, the four desirable properties do not guarantee a minimal number of constraint checks. We therefore refute the minimality claim in the paper introducing these properties. Furthermore, we propose a new desirable property for AC computation and extend AC6++ and AC7 to consider such a property. We show theoretically and experimentally that the new property provides a further substantial reduction in the number of constraint checks. 1