Sparse Constraint Graphs and Exceptionally Hard Problems

Many types of problem 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, occasional exceptionally hard problems (ehps) can be found in the easy and soluble region: these problems can be much harder than any problem occurring in the phase transition. We show that in binary constraint satisfaction problems ehps are much more likely to occur when the constraints are sparse than in dense problems. In ehps, the search algorithm encounters a large insoluble subproblem at an early stage; the exceptional di culty is due to the cost of searching the subproblem to prove insolubility. This cost can be dramatically reduced by using con ict-directed backjumping (CBJ) rather than a chronological backtracker. However, when used with forward checking and the failrst heuristic, it is only on hard problems occurring in the easy region that CBJ gives great savings over backtracking chronologically.