Edge-Directions of Standard Polyhedra with Applications to Network Flows

Recent results show that edge-directions of polyhedra play an important role in (combinatorial) optimization; in particular, a d-dimensional polyhedron with |D| distinct edge-directions has at most O(|D|^d-1) vertices. Here, we obtain a characterization of the directions of edges that are adjacent to a given vertex of a standard polyhedron of the form P = {x : Ax = b, l⩽ x⩽ u, tightening a standard necessary condition which asserts that such directions must be minimal support solutions of the homogenous equation Ax = 0 which are feasible at the given vertex. We specialize the characterization for polyhedra that correspond to network flows, obtaining a graph characterization of circuits which correspond to edge-directions. Applications to partitioning polyhedra are discussed. Research of these authors was supported by a grant from ISF – the Israel Science Foundation, by a VPR grant at the Technion, and by the Fund for the Promotion of Research at the Technion.