Euclidean distortion and the sparsest cut

Abstract

We prove that every n n -point metric space of negative type (and, in particular, every n n -point subset of L 1 L_1 ) embeds into a Euclidean space with distortion O ( log ⁡ n ⋅ log ⁡ log ⁡ n ) O(\sqrt {\log n} \cdot \log \log n) , a result which is tight up to the iterated logarithm factor. As a consequence, we obtain the best known polynomial-time approximation algorithm for the Sparsest Cut problem with general demands. If the demand is supported on a subset of size k k , we achieve an approximation ratio of O ( log ⁡ k ⋅ log ⁡ log ⁡ k ) O(\sqrt {\log k}\cdot \log \log k) .

Hard Metrics from Cayley Graphs of Abelian Groups

“…We say that a metric space V = (X, d) is of negative type if there is a map f from X to an Euclidean space such that for all x, y ∈ X we have d(x, y) = f (x) − f (y) 2 2 . (So all triangles spanned by the image of X are acute.)…”

“…When studying a special class of metric spaces, perhaps the most natural first question is whether this class contains hard metrics with respect to 2 . Many fundamental results in the modern theory of finite metric spaces may be viewed as a negative answer to this question for some special important class of metrics.…”

On the Advantage of Overlapping Clusters for Minimizing Conductance

“…The [26] algorithm is also the only non-trivial combinatorial approximation algorithm for Directed multicut. For more papers on cut problems see [27,6,15,13,35,10]. Some of these papers were an inspiration for this paper.…”

On the Unique Games Conjecture

“…Clearly, OPT(I) ≤ SDP(I) ≤ α C · OPT(I), and hence the SDP gives an α C approximation to the CSP. 8 The formal statement of Raghavendra's result is:…”

“…In a breakthrough work, Arora, Rao, and Vazirani [10] showed that the SDP algorithm gives O( √ log N ) approximation to the Sparsest Cut problem without demands. This was extended to the demands version of the problem by Arora, Lee, and Naor [8], albeit with a slight loss in the approximation factor. As discussed, the result is equivalent to an upper bound on c 1 (NEG, N ).…”