On the geometry of graphs with a forbidden minor

We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the Gupta-Newman-Rabinovich-Sinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)approximate multi-commodity max-flow/min-cut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in every family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest. ? Every metric on a graph of pathwidth k embeds into a distribution over trees with distortion depending only on k. This is equivalent to the statement that any family of graphs excluding a fixed tree embeds into a distribution over trees with O(1) distortion. For graphs of treewidth k, GNRS showed that this is impossible even for k = 2. In particular, our result implies that pathwidth-k metrics embed into L1 wi...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: James R. Lee, Anastasios Sidiropoulos
• Keywords: Algorithms, GNRS Embedding Conjecture, Planar Embedding Conjecture, Planar Graphs, STOC 2009