Faster Algorithms for Maximum Flow in Directed Planar Graphs with Apices or with Vertex Capacities

A $k$-apex graph is a graph with a subset of $k$ vertices, called apices, whose removal makes the graph planar. We present an $O(k^2 n \log^3 n)$-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 [Borradaile et al., FOCS 2012, SICOMP 2017]. 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 $O(k^3 n \ \textrm{polylog}(nC))$-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by $C$, and $k$ sources and sinks. This improves by a factor of $k^2$ over the fastest algorithm previously known for this problem [Wang, SODA 2019].