Improved Throughput for All-or-Nothing Multicommodity Flows With Arbitrary Demands

Throughput is a main performance objective in communication networks. This paper considers a fundamental maximum throughput routing problem — the All-or-Nothing Multicommodity Flow (ANF) problem — in arbitrary directed graphs and in the practically relevant but challenging setting where demands can be (much) larger than the edge capacities, mandating the need for splittable flows (i.e., flows may not follow a single path). Formally, the input for the ANF problem is an edge-capacitated directed graph where we have a given number of source-destination node-pairs with their respective demands and strictly positive weights. The goal is to route a maximum weight subset of the given pairs (i.e., the weighted throughput), respecting the edge capacities: A commodity is routed if all of its demand is routed from its respective source to destination (this is the all-or-nothing aspect). We present a polynomial-time bi-criteria approximation randomized rounding framework for this NP-hard problem that yields an arbitrarily good approximation on the weighted throughput while violating the edge capacity constraints by at most a sublogarithmic multiplicative factor. We present two non-trivial linear programming relaxations that can be used in the framework; the first uses a novel edge-flow formulation and the second uses a packing formulation. We demonstrate the “equivalence” of these formulations and then highlight the advantages of each of the two approaches. We complement our theoretical results with a proof of concept empirical evaluation, considering a variety of network scenarios.