Conference Paper

Making AC-3 an Optimal Algorithm.

Conference: Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, IJCAI 2001, Seattle, Washington, USA, August 4-10, 2001
Source: DBLP

ABSTRACT The AC-3 algorithm is a basic and widely used arc consistency enforcing algorithm in Constraint Sat- isfaction Problems (CSP). Its strength lies in that it is simple, empirically efficient and extensible. However its worst case time complexity was not considered optimal since the first complexity result for AC-3 (Mackworth and Freuder, 1985) with the bound (ed3), where e is the number of constraints and d the size of the largest domain. In this paper, we show suprisingly that AC-3 achieves the opti- mal worst case time complexity with (ed2). The result is applied to obtain a path consistency algo- rithm which has the same time and space complex- ity as the best known theoretical results. Our exper- imental results show that the new approach to AC-3 is comparable to the traditional AC-3 implementa- tion for simpler problems where AC-3 is more effi- cient than other algorithms and significantly faster on hard instances. simplicity of arc revision in AC-3 makes it convenient for implementation and amenable to various extensions for many constraint systems. Thus while AC-3 is considered as being sub-optimal, it often is the algorithm of choice and can out- perform other theoretically optimal algorithms. In this paper, we show that AC-3 achieves worst case op- timal time complexity of (ed2). This result is surprising since AC-3 being a coarse grained "arc revision" algorithm (Mackworth, 1977), is considered to be non-optimal. The known results for optimal algorithms are all on fine grained "value revision" algorithms. Preliminary experiments show that the new AC-3 is comparable to the traditional implemen- tations on easy CSP instances where AC-3 is known to be substantially better than the optimal fine grained algorithms. In the hard problem instances such as those from the phase transition, the new AC-3 is significantly better and is compa- rable to the best known algorithms such as AC-6. We also show that the results for AC-3 can be applied immediately to obtain a path consistency algorithm which has the same time and space complexity as the best known theoretical results. 1

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