  The relationship between the sparsest cut and the maximum concurrent
multi-flow in graphs has been studied extensively. For general graphs with $k$
terminal pairs, the flow-cut gap is $O(\log k)$, and this is tight. But when
topological restrictions are placed on the flow network, the situation is far
less clear. In particular, it has been conjectured that the flow-cut gap in
planar networks is $O(1)$, while the known bounds place the gap somewhere
between $2$ (Lee and Raghavendra, 2003) and $O(\sqrt{\log k})$ (Rao, 1999).
  A seminal result of Okamura and Seymour (1981) shows that when all the
terminals of a planar network lie on a single face, the flow-cut gap is exactly
$1$. This setting can be generalized by considering planar networks where the
terminals lie on $\gamma>1$ faces in some fixed planar drawing. Lee and
Sidiropoulos (2009) proved that the flow-cut gap is bounded by a function of
$\gamma$, and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at
most $3\gamma$. We prove that the flow-cut gap is $O(\log\gamma)$, by showing
that the edge-weighted shortest-path metric induced on the terminals admits a
stochastic embedding into trees with distortion $O(\log\gamma)$, which is
tight.
  The preceding results refer to the setting of edge-capacitated networks. For
vertex-capacitated networks, it can be significantly more challenging to
control flow-cut gaps. While there is no exact vertex-capacitated version of
the Okamura-Seymour Theorem, an approximate version holds; Lee, Mendel, and
Moharrami (2015) showed that the vertex-capacitated flow-cut gap is $O(1)$ on
planar networks whose terminals lie on a single face. We prove that the
flow-cut gap is $O(\gamma)$ for vertex-capacitated instances when the terminals
lie on at most $\gamma$ faces. In fact, this result holds in the more general
setting of submodular vertex capacities.
