While it is known that using network coding can significantly improve the throughput of directed networks, it is a notorious open problem whether coding yields any advantage over the multicommodity flow (MCF) rate in undirected networks. It was conjectured by Li and Li (2004) that the answer is "no". In this talk, we show that even a small advantage over MCF can be amplified to yield a near-maximum possible gap. We prove that any undirected network with k source-sink pairs that exhibits a (1+eps) gap between its MCF rate and its network coding rate can be used to construct a family of graphs G whose gap is log(|G|)^c for some constant c<1. The resulting gap is close to the best currently known upper bound, log(|G|), which follows from the connection between MCF and sparsest cuts. 

Joint work with Mark Braverman and Ariel Schvartzman.

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