Approximated Consistency for the Automatic Recording Problem

In constraint optimization, global constraints play a decisive role. To develop an efficient optimization tool, we need to be able to assess whether we are still able to improve the objective function further. This observation has lead to the development of a special kind of global constraints, so-called optimization constraints [2, 5]. Roughly speaking, an optimization constraint expresses our wish to search for improving solutions only while enforcing feasibility for at least one of the constraints of the problem. Since optimization constraints essentially evolve as a conjunction of a constraint on the objective value and some constraint of the constraint program, for many optimization constraints achieving generalized arc-consistency turns out to be NP-hard. Consequently, weaker notions of consistency have been developed with the aim to get ourselves back into the realm of tractable inference techniques. In [6, 7], we introduced the concept of approximated consistency which is a refined and stronger notion of relaxed consistency [1] for optimization constraints. Approximated consistency asks that all assignments are removed from consideration whose commitment would cause a bound with guaranteed accuracy to drop below the given threshold. We study the automatic recording problem (ARP) that consists in the solution of a knapsack problem where items are associated with time intervals and only items can be selected whose corresponding intervals do not overlap. The combination of a knapsack constraint with non-overlapping time-interval constraints can be identified as a subproblem in many more scheduling problems. For example, satellite scheduling can be viewed as a refinement of the automatic recording problem. Therefore, it is of general interest to study a global constraint that augments the knapsack constraint with timeinterval consistency of selected items. This idea gives raise to the Automatic Recording Constraint (ARC), which we want to study in this paper. Obviously, as an augmentation of the knapsack constraint, achieving generalized arc-consistency for the ARC is NP-hard. Consequently, we will develop a filtering algorithm for the constraint that does not guarantee backtrack-free search for the ARP, but that achieves at least approximated consistency with respect to bounds of arbitrary accuracy.