Trying Harder to Fail First

Variable ordering heuristics can have a profound e ect on the performance of backtracking search algorithms for constraint satisfaction problems. The smallest-remaining-domain heuristic is a commonlyused dynamic variable ordering heuristic, used in conjunction with algorithms such as forward checking which look ahead at the e ects of each variable instantiation on those variables not yet instantiated. This heuristic has been explained as an implementation of the failrst principle, stated by Haralick and Elliott [7], i.e. that the next variable selected should be the one which is most likely to result in an immediate failure. We calculate the probability that a variable will fail when using the forward checking algorithm to solve a class of binary CSPs. We derive a series of heuristics, starting with smallest-remaining-domain, based on increasingly accurate estimates of this probability, and predict that if the failrst principle is sound, the more accurate the estimate the better the performance should be. We describe experiments applying these heuristics, in conjunction with the forward checking algorithm, to a large set of randomly-generated problems from the same class. Our predictions are not borne out by the experimental results: putting more e ort into estimating the probability does not in general pay o . Our results thus refute the failrst principle and show that the success of smallest-remaining-domain and related heuristics must be explained in some other way.