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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 238 (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 ℓ1
 In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science
, 2005
"... In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 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 atl ..."
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Cited by 125 (10 self)
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In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 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 ℓ1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [19], 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 [20]. Though these hardness results depend on the UGC, the integrality gap instances rely “only ” on the PCP reductions 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 a (log log n) 1/6−δ integrality gap for (nonuniform) SPARSEST CUT and MINIMUM UNCUT, and an optimal integrality gap for MAXIMUM CUT. All our SDP solutions satisfy the socalled “triangle inequality ” constraints. This also shows, for the first time, that the triangle inequality constraints do not add any power to the GoemansWilliamson’s SDP relaxation of MAXIMUM CUT. The integrality gap for SPARSEST CUT immediately implies a lower bound for embedding negative type metrics into ℓ1. It also disproves the nonuniform version of Arora, Rao and Vazirani’s Conjecture [5], asserting that the integrality gap of the SPARSEST CUT SDP, with the triangle inequality constraints, is bounded from above by a constant.
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 93 (20 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
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 43 (9 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Euclidean distortion and the Sparsest Cut (Extended Abstract)
"... We prove that every npoint metric space of negative type (in particular, every npoint subset of L1) embeds into a Euclidean space with distortion O ( √ log n log log n), a result which is tight up to the O(log log n) factor. As a consequence, we obtain the best known polynomialtime approximation ..."
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Cited by 7 (1 self)
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We prove that every npoint metric space of negative type (in particular, every npoint subset of L1) embeds into a Euclidean space with distortion O ( √ log n log log n), a result which is tight up to the O(log log n) factor. As a consequence, we obtain the best known polynomialtime approximation algorithm for the Sparsest Cut problem with general demands. If the demand is supported on a subset of size k, we achieve an approximation ratio of O ( √ log k log log k).
QUASISYMMETRIC PARAMETERIZATIONS OF TWODIMENSIONAL METRIC SPACES
, 2007
"... This work would not have been possible without the aid of many people and institutions. My apologies to those whose contributions I may have forgotten; your assistance is recorded by the existence of this dissertation. ..."
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Cited by 1 (0 self)
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This work would not have been possible without the aid of many people and institutions. My apologies to those whose contributions I may have forgotten; your assistance is recorded by the existence of this dissertation.
5 Expander Flows, Geometric Embeddings and Graph
"... Abstract. We give a O ( √ log n)approximation algorithm for the SPARSEST CUT, EDGE EXPANSION, 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. C ..."
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Cited by 1 (0 self)
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Abstract. We give a O ( √ log n)approximation algorithm for the SPARSEST CUT, EDGE EXPANSION, 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 ℜd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “approximate certificate ” for a graph’s expansion, which involves embedding an nnode expander in it with appropriate dilation and congestion. We call this an expander flow.
We study pairs of metric spaces (X, dX) and (Y, dY) for which there
"... Abstract. Let A = (aij) ∈ Mn(R) be an n by n symmetric stochastic matrix. For p ∈ [1, ∞) and a metric space (X, dX), let γ(A, d p X) be the infimum over those γ ∈ (0, ∞] for which every x1,..., xn ∈ X satisfy 1 n2 n ∑ n∑ dX(xi, xj) p � γ n ∑ n∑ aijdX(xi, xj) n p. i=1 j=1 i=1 j=1 Thus γ(A, d p X) me ..."
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Abstract. Let A = (aij) ∈ Mn(R) be an n by n symmetric stochastic matrix. For p ∈ [1, ∞) and a metric space (X, dX), let γ(A, d p X) be the infimum over those γ ∈ (0, ∞] for which every x1,..., xn ∈ X satisfy 1 n2 n ∑ n∑ dX(xi, xj) p � γ n ∑ n∑ aijdX(xi, xj) n p. i=1 j=1 i=1 j=1 Thus γ(A, d p X) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel d p