Quadratic horizontally elastic not-first/not-last filtering algorithm for cumulative constraint

The not-first/not-last rule is a pendant of the edge finding rule, generally embedded in the cumulative constraint during constraint-based scheduling. It is combined with other filtering rules for more pruning of the tree search. In this paper, the Profile data structure in which tasks are scheduled in a horizontally elastic way is used to strengthen the classic not-first/not-last rule. Potential not-first task intervals are selected using criteria (specified later in the paper), and the Profile data structure is applied to selected task intervals. We prove that this new rule subsumes the classic not-first rule. A quadratic filtering algorithm is proposed for the new rule, thus improving the complexity of the horizontally elastic not-first/not-last algorithm from O(n3) to O(n2). The fixed part of external tasks that overlap with the selected task intervals is considered during the computation of the earliest completion time of task intervals. This improvement increases the filtering power of the algorithm while remaining quadratic. Experimental results, on a well-known suite of benchmark instances of Resource-Constrained Project Scheduling Problems (RCPSPs), show that the propounded algorithms are competitive with the state-of-the-art not-first algorithms in terms of tree search and running time reduction.