Criticality for multicommodity flows

For k ≥ 1, the k-commodity flow problem is, we are given k pairs of vertices in a graph G, and we ask whether there exist k flows in the graph, where • the ith flow is between the ith pair of vertices, and has total value one; and • for each edge e, the sum of absolute values of the flows along e is at most one. We prove that for all k there exists n(k) such that if G is connected, and contraction-minimal such that the k-commodity flow problem is infeasible (that is, minimal in the sense that contracting any edge makes the problem feasible) then |V (G)|+ |E(G)| ≤ n(k). For integers k, p ≥ 1, the (k, p)-commodity flow problem is as above, with the extra requirement that the flow value of each flow along each edge is a multiple of 1/p. We prove that if p > 1, and G is connected, and contraction-minimal such that the (k, p)-commodity flow problem is infeasible, then |V (G)| + |E(G)| ≤ n(k), with the same n(k) as before, independent of p. In contrast, when p = 1 there are arbitrarily large contraction-minimal instances, even when k = 2. We give some other corollaries of the same approach, including • a proof that for all k ≥ 0 there exists p > 0 such that every feasible k-commodity flow problem has a solution in which all flow values are multiples of 1/p, and • a very simple polynomial-time algorithm to solve the (k, p) multicommodity flow problem when p > 1.