Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Lectures on Discrete Geometry
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Nonembeddability theorems via Fourier analysis
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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Hard metrics are the class of extremal metrics with respect to embedding into Euclidean Spaces: their distortion is as bad as it possibly gets, which is Ω(log n). Besides being very interesting objects akin to expanders and good codes, with rich structure of independent interest, such metrics are important for obtaining lower bounds in Combinatorial Optimization, e.g., on the value of MinCut/MaxFlow ratio for multicommodity flows. For more than a decade, a single family of hard metrics was known (see [10,3]). Recently, a different such family was found (see [8]), causing a certain excitement among the researchers in the area. In this paper we present another construction of hard metrics, different from [10,3], and more general yet clearer and simpler than [8]. Our results naturally extend to NEG and to l1.