Backtracking Algorithms for Disjunctions of Temporal Constraints

We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 − y1 6 r1 ∨ · · · ∨ xn − yn 6 rn, where x1, . . . , xn, y1, . . . , yn are variables ranging over the real numbers, r1, . . . , rn are real constants, and n> 1. This is a wide class of temporal constraints that can be used to model a variety of problems in temporal reasoning, scheduling, planning, and temporal constraint databases. We have implemented several progressively more efficient algorithms for the consistency checking problem for this class of temporal constraints: backtracking, backjumping, three variations of forward checking, and forward checking with backjumping. We have partially ordered the above algorithms according to the number of visited search nodes and the number of performed consistency checks. Although our problem is non-binary, our results agree with the results of Kondrak and van Beek who consider only binary constraints. We have also studied the performance of the above algorithms experimentally using randomly generated sets of data and job shop scheduling problems. The experiments with random instances allowed us to locate the hard region for this class of problems. The results show that hard problems occur at a critical value of the ratio of disjunctions to variables.  2000 Elsevier Science B.V. All rights reserved.