On-line algorithms for networks of temporal constraints

We consider a semi-dynamic setting for the Temporal Constraint Satisfaction Problem, where we are requested to maintain the path-consistency of a network under a sequence of insertions of new (further) constraints between pairs of variables. We show how to maintain path-consistent a network in the defined setting in O(nR3) amortized time on a sequence of Θ(n2) insertions, where n is the number of vertices of the network and R is its range, defined as the maximum size of the minimum interval containing all the intervals of a single constraint. Furthermore we extend our algorithms to deal with more general temporal networks where variables can be points and/or intervals and constraints can be also defined on pairs of variables of different kind. For such cases our algorithms maintain their performance. Finally we adapt our algorithms for maintaining also the arc-consistency of such general networks, which is a particular kind of path-consistency limited to paths of length 1. The property is maintained in O(R) amortized time for Θ(n2) insertions. In case of constraints consisting of simple intervals the algorithm also gives a solution to the satisfaction problem