Before revision: The most efficient algorithm currently known for computing maximum integer flows in planar graphs with vertex capacities and multiple sources and sinks Wang, SODA 2019 runs in O(k ^5n \text{ polylog ] O(k ^4n \text{ polylog ]
After revision: The most efficient algorithm currently known for computing maximum integer flows in planar graphs with vertex capacities and multiple sources and sinks Wang, SODA 2019 runs in O(k ^2 n \log^3 n)-time algorithm that, given a directed k-apex graph G with n vertices, arc capacities, a set of sources S and a set of sinks T, computes a maximum flow from S to T in G. This improves by a factor of k on the fastest algorithm previously known for this problem ] O(k ^4n \text{ polylog ]