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 1|prec|Σ_jw_jC_j 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 that 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 finest 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.