Near-optimal solutions and large integrality gaps for almost all instances of single-machine precedence-constrained scheduling

We consider the problem of minimizing the weighted sum of completion times on a single machine subject to bipartite precedence constraints in which all minimal jobs have unit processing time and zero weight, and all maximal jobs have zero processing time and unit weight. For various probability distributions over these instances-including the uniform distribution-we show several "almost all"-type results. First, we show that almost all instances are prime with respect to a well-studied decomposition for this scheduling problem. Second, we show that for almost all instances, every feasible schedule is arbitrarily close to optimal. Finally, for almost all instances, we give a lower bound on the integrality gap of various linear programming relaxations of this problem.