On multiplicative λ-approximations and some geometric applications

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Let F be a set system over an underlying finite set X, and let μ be a nonnegative measure over X. I.e., for every S ⊆ X, μ(S) = ∑x∈S μ(x). A measure μ*on X is called a multiplicative λ-approximation of μ on (F, X) if for every S ∈ F it holds that aμ(S) ≤ μ*(S) ≤ bμ(S), and b/a = λ ≥ 1. The central question raised and partially answered in the present paper is about the existence of meaningful structural properties of F implying that for any μ on X there exists an 1+ε/1-ε-approximation μ*supported on a small subset of X. It turns out that the parameter that governs the support size of a multiplicative approximation is the triangular rank of F, trk(F). It is defined as the maximal length of a sequence of sets {Si}i=1t in T such that for all 1 < i ≤ t, Si 2 Uj < i Sj. We show that for any μ on X and 0 < ε < 1, there is measure μ*that 1+ε/1-ε-approximates μ on (X, F), and has support of size O(trk(F)2 log(trk(F))/poly(ε)). We also present two alternative constructions which in some cases improve upon this bound. Conversely, we show that for any 0 ≤ ε < 1 there exists a μ on X that cannot be 1+ε/1-ε approximated on (F, X) by any μ*with support of size < trk(F). For special families F this bound can be improved to Ω(trk(F)/ε). As an application we show a new dimension-reduction result for ℓ1 metrics: Any ℓ1-metric on n points can be (efficiently) embedded with 1+ε/1-ε-distortion into ℝO(n/ε2) equipped with the ℓ1 norm. This improves over the best previously known bound of O(n log n/poly(ε)) on dimension, due to Schechtman. We obtain also some new results on efficient sampling of Euclidean volumes. In order to make the general framework applicable to this setting, we develop the basic theory of finite volumes, analogous to the theory of finite metrics, and get results of independent interest in this direction. To do so, we use basic combinatorial/topological facts about simplicial complexes, and study the naturally arising questions.

Original languageEnglish
Title of host publicationProceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
PublisherAssociation for Computing Machinery
Number of pages17
ISBN (Print)9781611972108
StatePublished - 2012
Event23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan
Duration: 17 Jan 201219 Jan 2012

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

ASJC Scopus subject areas

  • Software
  • General Mathematics


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