  We prove that, for every $k=1,2,...,$ every shortest-path metric on a graph
of pathwidth $k$ embeds into a distribution over random trees with distortion
at most $c$ for some $c=c(k)$. A well-known conjecture of Gupta, Newman,
Rabinovich, and Sinclair states that for every minor-closed family of graphs
$F$, there is a constant $c(F)$ such that the multi-commodity max-flow/min-cut
gap for every flow instance on a graph from $F$ is at most $c(F)$. The
preceding embedding theorem is used to prove this conjecture whenever the
family $F$ does not contain all trees.
