# Pathwidth, trees, and random embeddings - Mathematics > Metric Geometry

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Abstract: We prove that, for every $k=1,2,

.,$ every shortest-path metric on a graphof pathwidth $k$ embeds into a distribution over random trees with distortionat most $c$ for some $c=ck$. A well-known conjecture of Gupta, Newman,Rabinovich, and Sinclair states that for every minor-closed family of graphs$F$, there is a constant $cF$ such that the multi-commodity max-flow-min-cutgap for every flow instance on a graph from $F$ is at most $cF$. Thepreceding embedding theorem is used to prove this conjecture whenever thefamily $F$ does not contain all trees.

Author: ** James R. Lee, Anastasios Sidiropoulos**

Source: https://arxiv.org/