  The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a $1-\epsilon$
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
$1-\epsilon$ approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
$\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))$.
