The Phase Transition Behaviour of Maintaining Arc Consistency

In this paper, we study two recently presented algorithms employing a \full look-ahead" strategy: MAC (Maintaining Arc Consistency); and the hybrid MAC-CBJ, which combines con ict-directed backjumping capability with MAC. We observe their behaviour with respect to the phase transition properties of randomly-generated binary constraint satisfaction problems, and investigate the bene ts of maintaining a higher level of consistency during search by comparing MAC and MAC-CBJ with the FC and FC-CBJ algorithms, which maintain only node consistency. The phase transition behaviour that has been observed for many classes of problem as a control parameter is varied has prompted a urry of research activity in recent years. Studies of these transitions, from regions where most problems are easy and soluble to regions where most are easy but insoluble, have raised a number of important issues such as the phenomenon of exceptionally hard problems (\ehps") in the easy-soluble region, and the growing realisation that the position of a problem in relation to the phase transition may have a major e ect on the relative performance of di erent algorithms. We therefore apply the algorithms studied to problems covering a range of sizes, topologies and positions in relation to the phase transition, paying particular attention to performance in relation to the occurrence of ehps. It is shown that increasing look-ahead greatly reduces the areas of the search space which must be explored, enabling backtrack-free searches over a larger range of values of the control parameter, and also greatly reduces the occurrence of ehps. We also observe that the performance of MAC scales much better than that of FC as problem size increases. The addition of CBJ to MAC has little e ect on most problems, but further reduces the incidence of ehps to produce stable performance in almost all populations of problems.

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