  We give an $\widetilde{O}({m^{3/2 - 1/762} \log (U+W))}$ time algorithm for
minimum cost flow with capacities bounded by $U$ and costs bounded by $W$. For
sparse graphs with general capacities, this is the first algorithm to improve
over the $\widetilde{O}({m^{3/2} \log^{O(1)} (U+W)})$ running time obtained by
an appropriate instantiation of an interior point method [Daitch-Spielman,
2008].
  Our approach is extending the framework put forth in [Gao-Liu-Peng, 2021] for
computing the maximum flow in graphs with large capacities and, in particular,
demonstrates how to reduce the problem of computing an electrical flow with
general demands to the same problem on a sublinear-sized set of vertices --
even if the demand is supported on the entire graph. Along the way, we develop
new machinery to assess the importance of the graph's edges at each phase of
the interior point method optimization process. This capability relies on
establishing a new connections between the electrical flows arising inside that
optimization process and vertex distances in the corresponding effective
resistance metric.
