On the facial structure of scheduling polyhedra

A well-known job shop scheduling problem can be formulated as follows. Given a graph G with node set N and with directed and undirected ares, find an orientation of the undirected ares that minimizes the length of a longest path in G. We treat the problem as a disjunctive, program, without recourse to integer, variables, and give a partial characterization of the scheduling polyhedron P(N), i.e., the convex hull of feasible schedules. In particular, we derive all the facet inducing inequalities for the scheduling polyhedron P(K) defined on some clique with node set K, and give a sufficient condition, for such inequalities to also induce facets of P(N). One of our results is that any inequality that induces a facet of P(H) for some H⊂K, also induces a facet of P(K). Another one is a characterization of adjacent facets in terms of the index sets of the nonzero coefficients of their defining inequalities. We also address the constraint identification problem, and give a procedure for finding an inequality that cuts off a given solution to a subset of the constraints.