  In this paper we study the min-cost flow problem in planar networks. We start
with the min-cost flow problem and apply two transformations, one is based on
geometric duality of planar graphs and the other on linear programming duality.
The result is a min-cost flow problem in a related planar network whose balance
constraints are defined by the costs of the original problem and whose costs
are defined by the capacities of the original problem. We use this
transformation to show an O(n log^2 n) time algorithm for the min-cost flow
problem in an n-vertex outerplanar network.
