Single-Machine Scheduling Polyhedra with Precedence Constraints

We consider nonpreemptive single-machine scheduling subject to precedence constraints. We define feasible schedules by the vector of the job completion times and study the structure of the convex hull of all feasible schedules, called the scheduling polyhedron P. We derive classes of valid inequalities for P, and necessary and sufficient conditions under which they are facet-inducing. Our main result is a complete description of the minimal linear system defining P when the precedence constraints are series-parallel. Moreover, this system consists of two classes of facet-inducing inequalities, associated with series and parallel compositions, respectively. If the precedence constraints are not series-parallel, we present another class of facet-inducing inequalities for P, associated with induced Z-subgraphs. We also show that the convex hull of all preemptive feasible schedules is identical to P if and only if the precedence constraints form an out-forest.