Where the Exceptionally Hard Problems Are 1

Constraint satisfaction problems exhibit a phase transition as a problem parameter is varied, from a region where most problems are easy and soluble to a region where most problems are easy but insoluble. In the intervening phase transition region, the median problem di culty is greatest. However, in the easy and soluble region, occasional exceptionally hard problems (ehps) can be found, which can be much harder than any problem occurring in the phase transition. In such problems, the rst few assignments made by the algorithm create an insoluble subproblem, but the algorithm cannot detect that there is no solution except by an exhaustive search. While searching the subproblem, the algorithm thrashes, repeatedly discovering the same inconsistency. Di erent algorithms have di erent susceptibilities to ehps, because some algorithms are better than others at recognising that the subproblem has no solution. We investigate the occurrence of ehps in random binary constraint satisfaction problems, using several constraint satisfaction algorithms, with a range of lookahead capabilities and combined with either chronological backtracking or con ict-directed backjumping. This leads to new insights into the behaviour of these algorithms and into the phenomenon of thrashing.