Efficient management and propagation of temporal constraints is important for temporal planning as well as for scheduling. During plan development, new events and temporal constraints are added and existing constraints may be tightened; the consistency of the whole temporal network is frequently checked; and results of constraint propagation guide further search. Recent work shows that enforcing partial path consistency provides an efficient means of propagating temporal information for the popular Simple Temporal Network (STN). We show that partial path consistency can be enforced incrementally, thus exploiting the similarities of the constraint network between subsequent edge tightenings. We prove that the worst-case time complexity of our algorithm can be bounded both by the number of edges in the chordal graph (which is better than the previous bound of the number of vertices squared), and by the degree of the chordal graph times the number of vertices incident on updated edges. We show that for many sparse graphs, the latter bound is better than that of the previously best-known approaches. In addition, our algorithm requires space only linear in the number of edges of the chordal graph, whereas earlier work uses space quadratic in the number of vertices. Finally, empirical results show that when incrementally solving sparse STNs, stemming from problems such as Hierarchical Task Network planning, our approach outperforms extant algorithms. @en

The Simple Temporal Problem (STP) is a sub-problem of almost any planning or scheduling problem involving time constraints. An existing efficient method to solve the STP, called ΔSTP, is based on partial path consistency and starts from a chordal constraint graph. In this paper, we analyse this algorithm and show that there exist instances for which its time complexity is quadratic in the number of triangles in the constraint graph. We propose a new algorithm, P³C, whose worst-case time complexity is linear in the number of triangles. We show both formally and experimentally that P³C outperforms ΔSTP significantly.

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