Polyhedra related to undirected multicommodity flows

Abstract Let G= (VG, EG) and H= (VH, EH) be two undirected graphs, and VH ⊆ VG. We associate with G and H the (unbounded) polyhedron P(G,H) in Q EG which consists of all nonegative rational-valued functions (vectors) l on EG such that, for each edge st in H, the distance between s and t in the graph G whose edges ρ ∈ EG have the lengths l(ρ) is no less that 1. Let ν(h) be the least positive integer k such that each vertex of P(G, H) is 1/k-integral for any G with VG⊇VH [ν(H)=∞ if such a k does not exist]. In other terms, ν(H) is the least positive integer k such that each problem dual to a maximum undirected multicommodity flow problem with the “commodity graph” H has on optimal solution that is 1/k-integral. We prove that νH) can be only 1,2,4,∞, and moreover, for each k = 1,2,4,∞, we describe the class of H's with ν(H)=k. Also results concerning extreme of cones related to feasibility multicommodity flow problems are presented.