Coarse Differentiation and Multi-flows in Planar Graphs

Abstract

We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound [8] for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair.This also improves the largest known gap for planar graphs from 3 2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the coarse differentiation method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into L 1 .

A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

“…Their first usage in Banach space theory is apparently due to James [18], and in non-linear setting to Johnson and Schechtman [20]. This argument is by now standard, Lee and Raghavendra [26,Lemma 4.1] prove essentially the same lemma as ours, but since their terminology is different, we decided to enclose the following elementary proof for convenience of the readers.…”

“…Lee and Raghavendra [26] proved that 2 is best possible -it is the supremum of 1 -distortions of the family of all multibranching diamonds D n,k , for all n, k ∈ N, with uniform weights on all edges, that is, the same family of graphs that we study in Theorem 1.3.…”

“…Multibranching diamonds are a generalization of usual (binary) diamond graphs. Their embedding properties were first studied in [26]. Definition 1.2 (cf.…”

“…Also, in recent years diamond graphs of high branching have appeared naturally in different contexts, cf. [4,26,43].…”

On the geometry of graphs with a forbidden minor

“…In [11], it is proved that c1(Treewidth(2)) < ∞, and later works [16,5] nailed down the precise dependence c 1 (Treewidth(2)) = 2. Extending such a bound even to Treewidth(3) seems quite difficult, and is a well-known open problem.…”