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:  Wang, SODA 2019 runs in O(k ^5n \text{ polylog ] O(k ^4n \text{ polylog ]