Improved timetable edge finder rule for cumulative constraint with profile

Abstract

Data structures are the main ingredient to strengthen both the time complexity and the filtering power of algorithms in Constraint-Based Scheduling. The TimeTable and the Profile are well-known data structures applied to filtering algorithms for cumulative constraint. The two data structures in this paper are applied simultaneously to overload checking and edge-finding rules. The resulting rules named TimeTable Horizontally Elastic Overload Checker and TimeTable Horizontally Elastic Edge Finder rules respectively subsume the enhancement of the overload checking rule and the edge-finding rule with the individual data structure. This new edge-finding rule is relaxed after a successive application of Profile on well-selected task intervals, then TimeTable on the new horizontally elastic edge-finding rule. Potential task intervals for the edge-finding rule are selected based on two criteria (specified later in the paper) and the strong detection rule of the horizontally elastic edge finder rule of Fetgo Betmbe and Djamegni (2022) is then applied to those selected task intervals. The new horizontally elastic edge-finder rule subsumes the edge-finding rule and is not comparable to the timetable edge-finding rule. A two-phase filtering algorithm of complexity $O\left(n^2\right)$ each (where $n$ is the number of tasks sharing the resource) is proposed for the new rule. Improvements based on the TimeTable are obtained by considering fixed parts of external tasks that overlap with the potential task intervals. The improved rule subsumes the timetable edge-finding rule, and a quadratic algorithm is derived from the previous algorithm. Experimental results, on a well-known suite of benchmark instances of Resource-Constrained Project Scheduling Problems, show that the propounded algorithms are competitive with the state-of-the-art algorithms regarding running time and tree search reduction.