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49
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.
The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into `1
 In Proc. 46th IEEE Symp. on Foundations of Comp. Sci
, 2005
"... In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an npoint negative type metric which requires distortion ..."
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Cited by 170 (11 self)
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In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an npoint negative type metric which requires distortion atleast (log log n)1/6−δ to embed into `1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [20], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies superconstant hardness results for (nonuniform) Sparsest Cut and Minimum Uncut problems. It is already known that the UGC also implies an optimal hardness result for Maximum Cut [21]. Though these hardness results rely on the UGC, we demonstrate, nevertheless, that the corresponding PCP reductions can be used to construct “integrality gap instances ” for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then, we “simulate ” the PCP reduction, and “translate ” the integrality gap instance of Unique Games to integrality gap instances for the respective cut problems! This enables us to prove
The multiplicative weights update method: a meta algorithm and applications
, 2005
"... Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies ..."
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Cited by 147 (13 self)
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Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1
Euclidean distortion and the Sparsest Cut
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] ..."
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Cited by 113 (22 self)
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BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every
Improved approximation algorithms for minimum weight vertex separators
 In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS’89
, 1989
"... vertex separators ..."
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Fast Algorithms for Approximate Semidefinite Programming using the Multiplicative Weights Update Method
"... Semidefinite programming (SDP) relaxations appear inmany recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangianrelaxation based technique (modified from the papers of Plotkin, Shmoys,and Tardos (PST), and ..."
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Cited by 41 (6 self)
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Semidefinite programming (SDP) relaxations appear inmany recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangianrelaxation based technique (modified from the papers of Plotkin, Shmoys,and Tardos (PST), and Klein and Lu) to derive faster algorithms for approximately solving several families of SDPrelaxations. The algorithms are based upon some improvements to the PST ideas which lead to new results even fortheir framework as well as improvements in approximate eigenvalue computations by using random sampling.
Improved lower bounds for embeddings into L1
 SIAM J. COMPUT.
, 2009
"... We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bo ..."
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Cited by 39 (5 self)
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We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an npoint metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a wellknown semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) 1/6−o(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l1, in Proceedings of the 46th Annual IEEE Symposium
Efficient Algorithms Using The Multiplicative Weights Update Method
, 2006
"... Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more eff ..."
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Cited by 28 (1 self)
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Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In this thesis, we explore applications of the Multiplicative Weights method in the design of efficient algorithms for various optimization problems. This method, which was repeatedly discovered in quite diverse fields, is an algorithmic technique which maintains a distribution on a certain set of interest, and updates it iteratively by multiplying the probability mass of elements by suitably chosen factors based on feedback obtained by running another algorithm on the distribution. We present a single metaalgorithm which unifies all known applications of this method in a common framework. Next, we generalize the method to the setting of symmetric matrices rather than real numbers. We derive the following applications of the resulting Matrix Multiplicative Weights algorithm: 1. The first truly general, combinatorial, primaldual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for obtaining O(plog n) approximations to various graph partitioning problems, such as Sparsest Cut, Balanced Separator in both directed and undirected weighted graphs, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the AlonRoichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3) time deterministic O(log n) approximation algorithm for the quantum hypergraph covering problem. 4. An alternative proof of a result of Aaronson that the flfatshattering dimension of quantum states on n qubits is O ( nfl2).
Ramsey partitions and proximity data structures
 J. European Math. Soc 9
"... This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The nonlinear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric sp ..."
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Cited by 26 (3 self)
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This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The nonlinear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the nonlinear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for everyε∈(0, 1), any npoint metric space has a subset of size n 1−ε which embeds into Hilbert space with distortion O(1/ε). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design the best known approximate distance oracles when the distortion is large, closing a gap left open by Thorup and Zwick in [31]. Namely, we show that for any n point metric space X, and k≥1, there exists an O(k)approximate distance oracle whose storage requirement is O ( n 1+1/k) , and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.
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.