The Satis ability Constraint

We describe an experimental investigation of the satissability phase transition for several diierent classes of randomly generated problems. We show that the \conventional" picture of easy-hard-easy problem diiculty is inadequate. In particular, there is a region of very variable problem dii-culty where problems are typically underconstrained and satissable. Within this region, problems can be orders of magnitude harder than problems in the middle of the satissability phase transition. These extraordinary hard problems appear to be associated with a constraint gap, a minimum in the amount of constraint propagation compared to the amount of search. We show that the position and shape of this constraint gap are very consistent with problem size. Unlike hard problems in the middle of satissability phase transition, hard problems in the variable region are not critically constrained between satissability and unsatissability. Indeed, hard problems in the variable region often contain a small and unique minimal unsatissable subset or reduce at an early stage in search to a hard unsatissable subprob-lem with a small and unique minimal unsatissable subset. The diiculty in solving such problems is thus in identifying the minimal unsatissable subset from the many irrelevant clauses. The existence of a constraint gap greatly hinders our ability to nd such minimal unsatissable subsets. We conjecture that these results will generalise both to other SAT problem classes, and to the phase transitions of other NP-hard problems.