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 ^5n \text{ polylog ]. Our improvement is achieved by introducing a new variant of the push-relabel algorithm for computing maximum flows. We use our improved algorithm for maximum flow in k-apex graphs, together with additional insights, to obtain an O(k ^4n \text{ polylog ]