Max Flows in Planar Graphs with Vertex Capacities

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 {k2 n, n log3 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 linear time when k is fixed but larger than 2. The second algorithm runs in O(k5 Δ n polylog (nU)) time, where each arc capacity and finite vertex capacity is bounded by U, and Δ is the maximum degree of the graph. Finally, when k = 3, we present an algorithm that runs in O(n log n) time; this algorithm works even when the capacities are arbitrary reals. Our algorithms improve on the fastest previously known algorithms when k and Δ are fixed and U is bounded by a polynomial in n. Prior to this result, the fastest algorithms ran in O(n4/3+o(1)) time for unit capacities; in the smallest of O(n3/2log n log U), Õ(n10/7U1/7), O(n11/8+o(1)U1/4), and O(n4/3 + o(1)U1/3) time for integer capacities; and in O(n2/log n) time for real capacities, even when k = 3.