Pathwidth, trees, and random embeddings

Abstract

We prove that, for every integer k≥1, every shortest-path metric on a graph of pathwidth k embeds into a distribution over random trees with distortion at most c=c(k), independent of the graph size. A well-known conjecture of Gupta, Newman, Rabinovich, and Sinclair [12] 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.

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Correspondence to James R. Lee.

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A portion of the results in this paper were announced at the 41st Annual Symposium on the Theory of Computing [19].

Research partially supported by NSF grant CCF-0644037 and a Sloan Research Fellowship.

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Lee, J.R., Sidiropoulos, A. Pathwidth, trees, and random embeddings. Combinatorica 33, 349–374 (2013). https://doi.org/10.1007/s00493-013-2685-8

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Mathematics Subject Classification (2010)

  • 05C12
  • 05C21