We present fully polynomial approximation schemes for concurrent multicommodity flow problems that run in time of the minimum possible dependencies on the number of commodities k. We show that by modifying the algorithms by Garg andKönemann [1998] andFleischer [2000], we can reduce their running time on a graph with n vertices and m edges fromÕ(ε −2 (m 2 + km)) toÕ(ε −2 m 2 ) for an implicit representation of the output, orÕ(ε −2 (m 2 + kn)) for an explicit representation, wherẽThe implicit representation consists of a set of trees rooted at sources (there can be more than one tree per source), and with sinks as their leaves, together with flow values for the flow directed from the source to the sinks in a particular tree. Given this implicit representation, the approximate value of the concurrent flow is known, but if we want the explicit flow per commodity per edge, we would have to combine all these trees together, and the cost of doing so may be prohibitive. In case we want to calculate explicitly the solution flow, we modify our schemes so that they run in time polylogarithmic in nk (n is the number of nodes in the network). This is within a polylogarithmic factor of the trivial lower bound of time (nk) needed to explicitly write down a multicommodity flow of k commodities in a network of n nodes. Therefore our schemes are within a polylogarithmic factor of the minimum possible dependencies of the running time on the number of commodities k. ACM Reference Format:Karakostas, G. 2008. Faster approximation schemes for fractional multicommodity flow problems.

We examine how the selfish behavior of heterogeneous users in a network can be regulated through economic disincentives, i.e., through the introduction of appropriate taxation. One wants to impose taxes on the edges so that any traffic equilibrium reached by the selfish users who are conscious of both the travel latencies and the taxes will minimize the social cost, i.e., will minimize the total latency. We generalize previous results of Cole, Dodis and Roughgarden that held for a single origin-destination pair to the multicommodity setting.Our approach, which could be of independent interest, is based on the formulation of traffic equilibria as a nonlinear complementarity problem by Aashtiani and Magnanti [1]. We extend this formulation so that each of its solutions will give us a set of taxes that forces the network users to conform, at equilibrium, to a certain prescribed routing. We use the special nature of the prescribed minimum-latency flow in order to reduce the difficult nonlinear complementarity formulation to a pair of primal-dual linear programs. LP duality is then enough to derive our results.

We reduce the approximation factor for Vertex Cover to 2 − Θ( 1 √ log n ) (instead of the previous 2 − Θ( log log n log n ), obtained by , and by Monien and Speckenmeyer [11]). The improvement of the vanishing factor comes as an application of the recent results of Arora, Rao, and Vazirani [2] that improved the approximation factor of the sparsest cut and balanced cut problems. In particular, we use the existence of two big and well-separated sets of nodes in the solution of the semidefinite relaxation for balanced cut, proven in [2]. We observe that a solution of the semidefinite relaxation for vertex cover, when strengthened with the triangle inequalities, can be transformed into a solution of a balanced cut problem, and therefore the existence of big well-separated sets in the sense of [2] translates into the existence of a big independent set.A preliminary version of this work appeared as a McMaster University Technical

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