Parameterized Analysis of a Dynamic Programming Algorithm for a Parallel Machine Scheduling Problem

We consider in this paper the scheduling problem defined by a set of dependent jobs with release times and deadlines to be processed by identical parallel machines. This problem is denoted by P|prec, ri, di|(*) in the literature. Starting from an extension of the Branch-and-Bound algorithm of Demeulemeester and Herroelen to take into account release times and deadlines, we build a state graph of which longest paths represent all active schedule. New dominance rules are also proposed. We establish that our state graph construction algorithm is fixed-parameter tractable. The two parameters are the pathwidth, which corresponds to the maximum number of overlapping jobs time windows and the maximum execution time of a job. The algorithm is experimented on random instances. These experiments show that the pathwidth is also a key factor of the practical complexity of the algorithm.