A Note on Maximum Integer Flows in Directed Planar Graphs with Vertex Capacities

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 $O(k^5n\text{ polylog}(nU))$ where $k$ is the number of sources and sinks, and $U$ is the largest capacity of a single vertex. In this work we give a faster implementation for a procedure used by Wang's algorithm, improving the overall running time of his algorithm to $O(k^4n\text{ polylog}(nU))$