On sparse spanners of weighted graphs
Discrete & Computational Geometry
Approximating the bandwidth via volume respecting embeddings
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Improved bounds on the sample complexity of learning
Journal of Computer and System Sciences
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On the impossibility of dimension reduction in l1
Journal of the ACM (JACM)
Topological characteristics of random triangulated surfaces
Random Structures & Algorithms
Homological Connectivity Of Random 2-Complexes
Combinatorica
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Near Optimal Dimensionality Reductions That Preserve Volumes
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
On Approximating the Average Distance Between Points
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Hardness of embedding simplicial complexes in Rd
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Homological connectivity of random k-dimensional complexes
Random Structures & Algorithms
Proceedings of the forty-first annual ACM symposium on Theory of computing
A unified framework for approximating and clustering data
Proceedings of the forty-third annual ACM symposium on Theory of computing
Minors in random and expanding hypergraphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric Approximation Algorithms
Geometric Approximation Algorithms
Near Linear Lower Bound for Dimension Reduction in L1
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
On laplacians of random complexes
Proceedings of the twenty-eighth annual symposium on Computational geometry
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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) = ΣxεS μ(x). A measure μ* on X is called a multiplicative λ-approximation of μ on (F, X) if for every S ε 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}ti=1 in F such that for all 1 i ≤ t, Si ⊈ ∪ji Sj. We show that for any μ on X and 0 X, F), and has support of size O(trk(F)2log(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 X that cannot be 1+ε/1-ε-approximated on (F, X) by any μ* with support of size 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.