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39
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 312 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
On the Hardness of Approximating Multicut and SparsestCut
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1. ..."
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Cited by 102 (5 self)
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We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.
Some Topics in Analysis of Boolean Functions
"... This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an exten ..."
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Cited by 44 (0 self)
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This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an extension of Chernoff/Hoeffding bounds to higherdegree polynomials; and, hardness for approximation algorithms.
Low distortion embeddings for edit distance
 In Proceedings of the Symposium on Theory of Computing
, 2005
"... We show that {0, 1} d endowed with edit distance embeds into ℓ1 with distortion 2 O( √ log d log log d). We further show efficient implementations of the embedding that yield solutions to various computational problems involving edit distance. These include sketching, communication complexity, neare ..."
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Cited by 27 (1 self)
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We show that {0, 1} d endowed with edit distance embeds into ℓ1 with distortion 2 O( √ log d log log d). We further show efficient implementations of the embedding that yield solutions to various computational problems involving edit distance. These include sketching, communication complexity, nearest neighbor search. For all these problems, we improve upon previous bounds. 1
The computational hardness of estimating edit distance
 In Proceedings of the Symposium on Foundations of Computer Science
, 2007
"... We prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. To the best of our knowledge, this is the first computational setting in which the complexity of computing the edit distance is provably ..."
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Cited by 24 (8 self)
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We prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. To the best of our knowledge, this is the first computational setting in which the complexity of computing the edit distance is provably larger than that of Hamming distance. Our lower bound exhibits a tradeoff between approximation and communication, asserting, for example, that protocols with O(1) bits of communication can only obtain approximation α ≥ Ω(log d / log log d), where d is the length of the input strings. This case of O(1) communication is of particular importance since it captures constantsize sketches as well as embeddings into spaces like L1 and squaredL2, two prevailing algorithmic approaches for dealing with edit distance. Furthermore, the bound holds not only for strings over alphabet Σ = {0, 1}, but also for strings that are permutations (aka the Ulam metric). Besides being applicable to a much richer class of algorithms than all previous results, our bounds are neartight in at least one case, namely of embedding permutations into L1. The proof uses a new technique, that relies on Fourier analysis in a rather elementary way. 1
Compression bounds for Lipschitz maps from the Heisenberg group to L1
, 2009
"... We prove a quantitative biLipschitz nonembedding theorem for the Heisenberg group with its CarnotCarathéodory metric and apply it to give a lower bound on the integrality gap of the GoemansLinial semidefinite relaxation of the Sparsest Cut problem. ..."
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Cited by 23 (11 self)
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We prove a quantitative biLipschitz nonembedding theorem for the Heisenberg group with its CarnotCarathéodory metric and apply it to give a lower bound on the integrality gap of the GoemansLinial semidefinite relaxation of the Sparsest Cut problem.
Earth Mover Distance over HighDimensional Spaces
, 2007
"... The Earth Mover Distance (EMD) between two equalsize sets of points in R d is defined to be the minimum cost of a bipartite matching between the two pointsets. It is a natural metric for comparing sets of features, and as such, it has received significant interest in computer vision. Motivated by re ..."
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Cited by 22 (8 self)
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The Earth Mover Distance (EMD) between two equalsize sets of points in R d is defined to be the minimum cost of a bipartite matching between the two pointsets. It is a natural metric for comparing sets of features, and as such, it has received significant interest in computer vision. Motivated by recent developments in that area, we address computational problems involving EMD over highdimensional pointsets. A natural approach is to embed the EMD metric into ℓ1, and use the algorithms designed for the latter space. However, Khot and Naor [KN06] show that any embedding of EMD over the ddimensional Hamming cube into ℓ1 must incur a distortion Ω(d), thus practically losing all distance information. We circumvent this roadblock by focusing on sets with cardinalities upperbounded by a parameter s, and achieve a distortion of only O(log s · log d). Since in applications the feature sets have bounded size, the resulting distortion is much smaller than the Ω(d) lower bound. Our approach is quite general and easily extends to EMD over R d. We then provide a strong lower bound on the multiround communication complexity of estimating EMD, which in particular strengthens the known nonembeddability result of [KN06]. Our bound exhibits a smooth tradeoff between approximation and communication, and for example implies that every algorithm that estimates EMD using constant size sketches can only achieve Ω(log s) approximation.
Partitioning Graphs into Balanced Components
, 2009
"... We consider the kbalanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components. We allow k to be part of the input and denote the cardinality of the verte ..."
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Cited by 22 (2 self)
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We consider the kbalanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components. We allow k to be part of the input and denote the cardinality of the vertex set by n. This problem is a natural and important generalization of wellknown graph partitioning problems, including minimum bisection and minimum balanced cut. We present a (bicriteria) approximation algorithm achieving an approximation of O ( √ log n log k), which matches or improves over previous algorithms for all relevant values of k. Our algorithm uses a semidefinite relaxation which combines ℓ 2 2 metrics with spreading metrics. Surprisingly, we show that the integrality gap of the semidefinite relaxation is Ω(log k) even for large values of k (e.g., k = n Ω(1)), implying that the dependence on k of the approximation factor is necessary. This is in contrast to previous approximation algorithms for kbalanced partitioning, which are based on linear programming relaxations and their approximation factor is independent of k.
Improved approximation of linear threshold functions
 In Proc. 24nd Annual IEEE Conference on Computational Complexity (CCC
, 2009
"... We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only ..."
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Cited by 19 (12 self)
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We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only on Inf(f) 2 · poly(1/ɛ) many variables, where Inf(f) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s wellknown theorem [Fri98], which states that every Boolean function f is ɛclose to a function depending only on 2 O(Inf(f)/ɛ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f) 2 + 1/ɛ 2) many variables are required for ɛapproximating threshold functions. Our second result is a proof that every nvariable threshold function is ɛclose to a threshold function with integer weights at most poly(n) · 2 Õ(1/ɛ2/3). This is an improvement, in the dependence on the error parameter ɛ, on an earlier result of [Ser07] which gave a poly(n) · 2 Õ(1/ɛ2) bound. Our improvement is obtained via a new proof technique that uses strong anticoncentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give lowweight approximators for threshold functions under a range of probability distributions other than the uniform distribution.