Maximum Integer Flows in Directed Planar Graphs with Vertex Capacities and Multiple Sources and Sinks

We consider the maximum flow problem in directed planar graphs with capacities on both vertices and arcs and with multiple sources and sinks. We present three algorithms when the capacities are integers. The first algorithm runs in O(min{kn logn, n log n+kn}) time when all capacities are bounded by a constant, where n is the number of vertices in the graph and k is the number of terminals. This algorithm is the first to solve the vertex-disjoint paths problem in nearlinear time when k is fixed but larger than 2. The second algorithm runs in O(kn polylog(nU)) time, where U is the largest finite capacity of a single vertex. Finally, when k = 3, we present an algorithm that runs in O(n logn) time; this algorithm works even when the capacities are arbitrary reals. Our algorithms improve on the fastest previously known algorithms when k is fixed and U is bounded by a polynomial in n. Prior to this result, the fastest algorithms ran in O(n/ logn) time for real capacities, O(n logn logU) for integer capacities, and Õ(n) for unit capacities, even