Efficient Algorithms for the Maximum Concurrent Flow Problem Pierre-Olivier Bauguion Télécom Sud Paris, Orange Labs., France Walid Ben-Ameur Télécom Sud Paris, CNRS Samovar UMR 5157, 9 rue Charles Fourier 91000 Evry, France Eric Gourdin Orange Labs., 38/40 rue du général leclerc 92130 Issy-Les-Moulineaux, France In this article, we propose a generic decomposition scheme for the maximum concurrent flow problem. This decomposition scheme encompasses many mod- els, including, among many others, the classical path formulation and the less studied tree formulation, where the flows of commodities sharing a same source vertex are routed on a set of trees. The pricing problem for this generic model is based on shortest-path computations. We show that the tree-based linear programming formula- tion can be solved much more quickly than the path or the aggregated arc-flow formulation. Some other decompo- sition schemes can lead to even faster resolution times. Finally, an efficient strongly polynomial-time combina- torial algorithm is proposed for the single-source case. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 65(1), 56–67 2015 Keywords: maximum concurrent flow; column generation; decompositions; combinatorial algorithm; shortest path; sparsest cut; tree-based formulation 1. INTRODUCTION Multicommodity flow models have been introduced long ago, as natural extensions of single flow problems, and, since then, have been extensively used in many contexts essentially to capture the movements of different types of commodities in various real-world activities such as shipment of goods in transportation models, routing of data streams in telecommu- nications networks, and distribution of energy, water flows, electricity (see, e.g., [1]). Received September 2014; accepted September 2014 Correspondence to: W. Ben-Ameur; e-mail: walid.benameur@telecom- sudparis.eu DOI 10.1002/net.21572 Published online 25 December 2014 in Wiley Online Library (wileyonlinelibrary.com). © 2014 Wiley Periodicals, Inc. In this article, we will focus on the maximum concur- rent flow (MCF) problem. We consider a directed graph with capacities on its arcs and a set of multicommodity demands to be routed through the graph. The objective is to com- pute the maximum γ such that a fraction γ of every demand can be simultaneously routed without exceeding the available capacities. Besides the applications previously mentioned, MCF is the dual of the linear programming (LP) relaxation of the sparsest cut problem. A sparsest cut is a cut minimizing the ratio of the capacity of the cut to the total demand that is separated by the cut. Many approximation algorithms for this problem are based on the solution of MCF (see, e.g., [26]). Another reason to consider MCF is that it gives rise to extremely difficult problems (see, e.g., [4, 5]). Even if MCF problems can be solved exactly in polynomial time using lin- ear programming, the computing time can be quite long. In fact, the MCF problem can even be solved in strongly poly- nomial time using the approach of [22], slightly improved in [15]. The approach is based on the strongly polynomial-time algorithm of [27] to solve linear programs where the size of the entries of the constraint matrix is polynomially bounded in the dimension of the problem. While the algorithms of [22] and [15] are nice theoretical contributions, they are not really of practical interest. This led many researchers to design approximation algorithms for MCF [5, 8, 9, 16, 17, 21]. Flows are traditionally modeled with arc-flow variables specifying the amount of each commodity on each arc. By considering classical flow conservation constraints and capacity constraints one can easily model MCF. However, such models become inefficient when the size of the linear program becomes huge. One way to reduce the size of the LP consists of aggregating the demands either by source or by destination (see [1, 11]). This generally allows one to solve larger instances. NETWORKS—2015—DOI 10.1002/net