The all-or-nothing multicommodity flow problem

We consider the all-or-nothing multicommodity flow problem in general graphs. We are given a capacitated undirected graph G = (V, E, u) and set of k pairs s₁t₁, s₂t₂..., s _kt_k. Each pair has a unit demand. The objective is to find a largest subset S of {1, 2,...,k} such that for every i in S we can send a flow of one unit between s_i and t_i. Note that this differs from the edge-disjoint path problem (EDP) in that we do not insist on integral flows for the pairs. This problem is NP-hard, and APX-hard, even on trees. For trees, a 2-approximation is known for the cardinality case and a 4-approximation for the weighted case. In this paper we build on a recent result of Räcke on low congestion oblivious routing in undirected graphs to obtain a poly-logarithmic approximation for the all-or-nothing problem in general undirected graphs. The best previous known approximation for all-or-nothing flow problem was O(min(n^2/3, √m)), the same as that for EDP. Our algorithm extends to the case where each pair s_it_i has a demand d_i associated with it and we need to completely route d _i to get credit for pair i. We also consider the online admission control version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it or not. We obtain a randomized algorithm with a competitive ratio that is similar to the approximation ratio for the offline algorithm.