Improved approximation algorithms for unsplittable flow problems

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Abstract
In the single-source unsplittable flow problem we are given a graph G, a source vertex s and a set of sinks t_1,\ldots,t_k with associated demands. We seek a single s-t_i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by its capacity c_e. The problem is an NP-hard variant of max flow and a generalization of single-source edge-disjoint paths with applications to scheduling, load balancing and virtual-circuit routing problems. In a significant development, Kleinberg gave recently constant-factor approximation algorithms for several natural optimization versions of the problem \cite{Kleinberg96}. In this paper we give a generic framework that yields simpler algorithms and significant improvements upon the constant factors. Our framework, with appropriate subroutines, applies to all optimization versions previously considered and treats in a unified manner directed and undirected graphs. To give a flavor of our results, consider minimizing relative congestion, i.e. the maximum ratio over all edges e of the flow through e divided by the capacity c_e. This metric was a primary testbed for randomized rounding techniques and has been studied extensively. We give a simple (3.23+o(1))-approximation algorithm for both directed and undirected graphs. The previously known bounds were 16 for the directed and 8.25 for the undirected case. Our approach also gives the first constant-factor approximation for minimum-cost unsplittable flow on directed graphs and improves considerably upon the approximation ratio for the minimum cost version on undirected graphs. The algorithmic techniques we introduce are quite general and apply to related problems as well. For example we use them to give a constructive proof of the following fact. If there exists an algorithm {\cal A} for (multisource) edge-disjoint paths such that {\cal A} outputs an approximation for relative congestion that is \rho times the fractional optimum, then a corresponding O(\rho)-approximation algorithm exists for multisource unsplittable flow with arbitrary demands and capacities. On the negative side we show that for the problem with two sources, no \rho-approximation with \rho0. We give a best possible, unless P=NP, 3/2-approximation for this restricted unsplittable flow problem and generalizations to other restricted sets of demands.

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