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Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows

The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a $1-ε$ approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem. As the underlying linear systems in the electrical problems of multicommodity flow problems are no longer Laplacians, our approach is tailored to generate specialized systems which can be preconditioned and solved efficiently using Laplacians. Given an undirected graph with m edges and k commodities, we give algorithms that find $1-ε$ approximate solutions to the maximum concurrent flow problem and the maximum weighted multicommodity flow problem in time $\tilde{O}(m^{4/3}\poly(k,ε^{-1}))$.