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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.

Flow-Cut Gaps and Face Covers in Planar Graphs