A binarized-domains arc-consistency algorithm for TCSPs: its computational analysis and its use as a filtering procedure in solution search algorithms

TCSPs (Temporal Constraint Satisfaction Problems), as defined in [Dechter et al., 1991], get rid of unary constraints by binarizing them after having added an "origin of the world" variable. In this work, we look at the constraints between the "origin of the world" variable and the other variables, as the (binarized) domains of these other variables. With this in mind, we define a notion of arc-consistency for TCSPs, which we will refer to as binarized-domains Arc-Consistency, or bdArc-Consistency for short. We provide an algorithm achieving bdArc-Consistency for a TCSP, which we will refer to as bdAC-3, for it is an adaptation of Mackworth's [1977] well-known arc-consistency algorithm AC-3. We show that if a convex TCSP, referred to in [Dechter et al., 1991] as an STP (Simple Temporal Problem), is bdArc-Consistent, and its "origin of the world" variable disconnected from none of the other variables, its binarized domains are minimal. We provide two polynomial backtrack-free procedures: one for the task of getting, from a bdArc-Consistent STP, either that it is inconsistent or, in case of consistency, a bdArc-Consistent STP refinement whose "origin of the world" variable is disconnected from none of the other variables; the other for the task of getting a solution from a bdArc-Consistent STP whose "origin of the world" variable is disconnected from none of the other variables. We then show how to use our results both in a general TCSP solver and in a TCSP-based job shop scheduler. From our work can be extracted a one-to-all all-to-one shortest paths algorithm of an IR-labelled directed graph. Finally, we show that an existing adaptation to TCSPs of Mackworth's [1977] path-consistency algorithm PC-2 is not guaranteed to always terminate, and correct it.