Minimal Consistency Problem of Temporal Qualitative Constraint Networks

Various formalisms for representing and reasoning about temporal information with qualitative constraints have been studied in the past three decades. The most known are definitely the Point Algebra (PA) and the Interval Algebra (IA) proposed by Allen. In this paper, for both calculi, we study a particular problem that we call minimal consistency problem (MinCons). Given a temporal qualitative constraint network (TQCN) and a positive integer k, this problem consists in deciding whether or not this TQCN admits a solution using at most k distinct points on the line. On the one hand, we prove that this problem is NP-complete for both PA and IA, in the general case. On the other hand, we show that for TQCNs defined on the convex relations, MinCons is polynomial. For these TQCNs, we give a polynomial method allowing to obtain compact scenarios.