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introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly ∆ 1−2/k colors. Here ∆ is the maximum degree in the graph and is assumed to be of the order of n δ for some 0 < δ < 1. Their results… (More)

MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an <italic>n</italic> dimensional sphere,… (More)

We show that any L 1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,. .. , n} 2 ⊆ R 2 incurs distortion Ω log n. We also use Fourier analytic techniques to construct a simple L 1 embedding of this space which has distortion O(log n).

—A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all… (More)

We consider the problem of embedding vectors from an arbitrary Euclidean space into a low dimensional Euclidean space, while preserving up to a small distortion, a subset of the distances. In particular, preserving only the distance of each vector to a small number of its nearest neighbors. We show that even when the subset of distances we wish to preserve… (More)