  We give an $O(k^3 n \log n \min(k,\log^2 n) \log^2(nC))$-time algorithm for
computing maximum integer flows in planar graphs with integer arc {\em and
vertex} capacities bounded by $C$, and $k$ sources and sinks. This improves by
a factor of $\max(k^2,k\log^2 n)$ over the fastest algorithm previously known
for this problem [Wang, SODA 2019].
  The speedup is obtained by two independent ideas. First we replace an
iterative procedure of Wang that uses $O(k)$ invocations of an $O(k^3 n \log^3
n)$-time algorithm for maximum flow algorithm in a planar graph with $k$ apices
[Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that
only makes one invocation of the algorithm of Borradaile et al. Second, we show
two alternatives for computing flows in the $k$-apex graphs that arise in our
modification of Wang's procedure faster than the algorithm of Borradaile et al.
In doing so, we introduce and analyze a sequential implementation of the
parallel highest-distance push-relabel algorithm of Goldberg and Tarjan~[JACM
1988].
