Modelling Exceptionally Hard Constraint Satisfaction Problems

Randomly-generated constraint satisfaction problems go through a phase transition as the constraint tightness varies. Loose constraints give an `easy-soluble' region, where problems have many solutions and are almost always easy to solve. However, in this region, systematic search algorithms may occasionally encounter problems which are extremely expensive to solve. It has been suggested that in these cases, the rst few instantiations made by the algorithm create an insoluble subproblem; an exhaustive search of the subproblem to prove its insolubility accounts for the high cost. We propose a model for the occurrence of such subproblems when using the backtracking algorithm. We calculate the probability of their occurrence and estimate their cost. From this, we derive the theoretical cost distribution when the constraint graph is complete and show that it matches the observed cost distribution in this region. We suggest that a similar model would also account for the exceptionally hard problems that have been observed using more sophisticated algorithms.