Embeddings of Topological Graphs: Lossy Invariants, Linearization, and 2-Sums

We study the properties of embeddings, multicommodity flows, and sparse cuts in minor-closed families of graphs which are also closed under 2-sums; this includes planar graphs, graphs of bounded treewidth, and constructions based on recursive edge replacement. In particular, we show the following. • Every graph which excludes K4 as a minor (in particular, series-parallel graphs) admits an embedding into L1 with distortion at most 2, confirming a conjecture of Gupta, Newman, Rabinovich, and Sinclair, and improving over their upper bound of 14. This shows that in every multi-commodity flow instance on such a graph, one can route a maximum concurrent flow whose value is at least half the cut bound. Our upper bound is optimal, as it matches a recent lower bound of Lee and Raghavendra. • We move beyond K4-minor-free graphs by showing that every W4-minor-free-graph embeds into L1 with O(1) distortion, where W4 is the 4-wheel. By a characterization of Seymour, these graphs are precis...

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Amit Chakrabarti, Alexander Jaffe, James R. Lee, Justin Vincent
• Keywords: FOCS 2008, K4-minor-free Graphs, Planar Graphs, Theoretical Computer Science, Upper Bound