Optimization in temporal qualitative constraint networks

Abstract

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 $$(\mathsf {PA})$$(PA) and the Interval Algebra $$({\mathsf {IA}})$$(IA) proposed by Allen. In this paper, for both calculi, we study a problem that we call the minimal consistency problem $$(\mathsf {MinCons})$$(MinCons). Given a temporal qualitative constraint network $$(\mathsf {TQCN})$$(TQCN) and a positive integer $$k$$k, this problem consists in deciding whether or not this $$\mathsf {TQCN}$$TQCN admits a solution using at most $$k$$k distinct points on the line.We show that $$\mathsf {MinCons}$$MinCons for $$\mathsf {PA}$$PA can be encoded into the finitary versions of Godel logic. Furthermore, we prove that the $$\mathsf {MinCons}$$MinCons problem is $$\mathsf {NP}$$NP-complete for both $$\mathsf {PA}$$PA and $${\mathsf {IA}}$$IA, in the general case. However, we show that for $$\mathsf {TQCN}$$TQCNs defined on the convex relations, $$\mathsf {MinCons}$$MinCons is polynomial. For such $$\mathsf {TQCN}$$TQCNs, we give a polynomial method that allows one to obtain compact scenarios.