Decompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling Problem

We present an in-depth theoretical, algorithmic and computational study of a linear programming (LP) relaxation to the precedence constrained single machine scheduling problem 1jprecj P j wjCj to minimize a weighted sum of job completion times. On the theoretical side, we study the structure of tight parallel inequalities in the LP relaxation and show that every permutation schedule which is consistent with Sidney's decomposition has total cost no more than twice the optimum. On the algorithmic side, we provide a parametric extension to Sidney's decomposition and show that a nest decomposition can be obtained by essentially solving a parametric minimum cut problem. Finally, we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2,000 jobs.