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 查看全文>>