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

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

Arc-consistency algorithms are the workhorse of backtrackers that Maintain Arc-Consistency (MAC). This report will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arc-consistency algorithm to have an optimal worst case time-complexity. To sacrifice this optimality allows MAC solvers that (1) do not need additional data structures during search, (2) have an excellent average time-complexity, and (3) have a space-complexity which improves significantly on that of MAC solvers that have optimal arc-consistency components. Results will be presented from an experimental comparison between MAC-2001, MAC-3 d and related algorithms. MAC-2001 has an arc-consistency component with an optimal worst case time-complexity, whereas MAC-3 d does not. MAC-2001 requires additional data structures during search, whereas MAC-3 d does not. MAC-3 d has a space-complexity 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 MAC-2001's space-complexity is O(ed min(n; d)). MAC-2001 required about 35% more solution time on average than MAC-3 d for easy and hard random problems. MAC-3 d recorded the least solution time for 21 of the 25 real-world problems. Our results indicate that if checks are cheap then lightweight algorithms like MAC-3 d are promising.

Abstract. Arc-consistency algorithms are the workhorse of backtrackers that Maintain Arc-Consistency (MAC). This paper will provide experimental evidence that, despite common belief to the contrary, it is not always necessary for a good arc-consistency algorithm to have an optimal worst case time-complexity. To sacrifice this optimality allows MAC solvers that (1) do not need additional data structures during search, (2) have an excellent average time-complexity, and (3) have a space-complexity which improves significantly on that of MAC solvers that have optimal arc-consistency components. Results will be presented from an experimental comparison between MAC-2001, MAC-3d and related algorithms. MAC-2001 has an arc-consistency component with an optimal worst case time-complexity, whereas MAC-3d does not. MAC-2001 requires additional data structures during search, whereas MAC-3d does not. MAC-3d has a space-complexity 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 MAC-2001’s space-complexity is O(ed min(n, d)). MAC-2001 required about 35 % more solution time on average than MAC-3d for easy and hard random problems, MAC-3d was faster for 40 % of the real-world problems but slower for the remaining real-world problems. Our results are an indication that if checks are cheap then lightweight algorithms like MAC-3d are promising. 1

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Maintaining arc consistency algorithms during the search with an optimal time and space complexity