Intelligent Backtracking on the Hardest Constraint Problems

A number of researchers have presented evidence that the hardest constraint satisfaction problems tend to arise near the \crossover" point, where around half the problems have solutions. Two recent papers , however, argue that while this is true of the median search cost, the mean search cost has an entirely diierent behavior. The mean search cost appears to be dominated by a relatively small number of extremely diicult problems, and these problems typically have constraint graphs that are much sparser than those of the problems near the crossover point. In this paper, we show that these outliers in the sparse region are not fundamentally diicult problems. They are hard for chronological backtracking only because of thrashing, and they are easily handled by more sophisticated search strategies such as backjumping and dependency-directed backtracking. Thus, they are quite diierent in character from the crossover-point problems, which appear to be diicult for all known search algorithms. We present our experimental results and discuss their implications.