Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in   Near-Linear Time
  We give an O(n log^3 n) algorithm that, given an n-node directed planar graph with arc capacities, a set of source nodes, and a set of sink nodes, finds a maximum flow from the sources to the sinks. Previously, the fastest algorithms known for this problem were those for general graphs. 
Glencora Borradaile, Philip N. Klein, Shay Mozes, Yahav Nussbaum,   Christian Wulff-Nilsen
cs.DM cs.DS