Coarse Differentiation and Multi-flows in Planar Graphs

We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound [8] for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair. This also improves the largest known gap for planar graphs from 32 to 2, yielding the first lower bound that doesn’t follow from elementary calculations. Our approach uses the coarse differentiation method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into L1.