On the convex hull of feasible solutions to certain combinatorial problems

Abstract

By Weyl's Theorem, the convex hull of the set of (characteristic vectors of) feasible solutions to combinatorial problems is a convex polyhedron when there is a finite number of feasible solutions. This need not be the case of infinite solution sets, even when they are finitely generated. We demonstrate this on examples of Steiner graphical travelling salesman problems, Steiner Chinese postman problems and single machine scheduling problems, either nonpreemptive with changeover times, or preemptive with deadlines or precedence constraints. We present a necessary and sufficient condition for the convex hull of a finite union of (unbounded) convex polyhedra to be closed, and hence a convex polyhedron itself. We apply this result to some of the above problems, and discuss conditions under which the convex hull of their feasible solutions is a convex polyhedron.