Lower Bounds for Embedding into Distributions over Excluded Minor Graph   Families
  It was shown recently by Fakcharoenphol et al that arbitrary finite metrics can be embedded into distributions over tree metrics with distortion O(log n). It is also known that this bound is tight since there are expander graphs which cannot be embedded into distributions over trees with better than Omega(log n) distortion.   We show that this same lower bound holds for embeddings into distributions over any minor excluded family. Given a family of graphs F which excludes minor M where |M|=k, we explicitly construct a family of graphs with treewidth-(k+1) which cannot be embedded into a distribution over F with better than Omega(log n) distortion. Thus, while these minor excluded families of graphs are more expressive than trees, they do not provide asymptotically better approximations in general. An important corollary of this is that graphs of treewidth-k cannot be embedded into distributions over graphs of treewidth-(k-3) with distortion less than Omega(log n).   We also extend a result of Alon et al by showing that for any k, planar graphs cannot be embedded into distributions over treewidth-k graphs with better than Omega(log n) distortion. 
Douglas E. Carroll and Ashish Goel