Semidefinite programming in combinatorial optimization
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Approximation algorithms
Lectures on Discrete Geometry
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On average distortion of embedding metrics into the line and into L1
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Cuts, Trees and -Embeddings of Graphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Metric Embeddings with Relaxed Guarantees
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
l22 spreading metrics for vertex ordering problems
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Volume distortion for subsets of Euclidean spaces: extended abstract
Proceedings of the twenty-second annual symposium on Computational geometry
A combinatorial, primal-dual approach to semidefinite programs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
An improved approximation ratio for the minimum linear arrangement problem
Information Processing Letters
Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
ACM Transactions on Algorithms (TALG)
Geometry, flows, and graph-partitioning algorithms
Communications of the ACM
Partitioning graphs into balanced components
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
On the Optimality of Gluing over Scales
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Bilipschitz snowflakes and metrics of negative type
Proceedings of the forty-second ACM symposium on Theory of computing
Genus and the geometry of the cut graph
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Near-optimal distortion bounds for embedding doubling spaces into L1
Proceedings of the forty-third annual ACM symposium on Theory of computing
A better approximation ratio for the vertex cover problem
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
On the hardness of embeddings between two finite metrics
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Dimensionality reduction: beyond the Johnson-Lindenstrauss bound
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Locally testable codes and cayley graphs
Proceedings of the 5th conference on Innovations in theoretical computer science
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A central open problem in the field of finite metric spaces is to find an efficient relaxation of the cut cone---the collection of positive linear combinations of cut pseudo-metrics on a finite set. In particular, it has been asked how well squared-Euclidean metrics (the so-called metrics of "negative type") embed into L1, and it is known that the answer to this question coincides with the integrality gap of a folklore semi-definite relaxation for computing the Sparsest Cut of a graph.Bourgain's classical embedding theorem implies that any n-point metric space embeds into L2 with O(log n) distortion. We give the first embeddings for metrics of negative type which beat Bourgain's bound. Specifically, we show that for every ∈ 0, there exists a δ 0 such that every n-point metric of negative type embeds into L2+∈, with distortion O(log n)1-δ. We also exhibit the first o(log n) bounds on the Euclidean distortion of finite subsets of Lp, for 1 p L1 and L2, and thus provide a necessary first step in resolving the long-standing open question on the Euclidean distortion of finite subsets of L1.In proving these results, we introduce a number of new techniques for the construction of low-distortion embeddings. These include a generic Gluing Lemma which avoids the overhead that typically arises from the naïve concatenation of different scales, and which provides new insights into the cut structure of finite graphs. We also exhibit the utility of Lipschitz extension theorems from Functional Analysis to the embedding of finite metric spaces. Finally, we prove the "Big Core" Theorem---a significantly improved and quantitatively optimal version of the main structural theorem in [ARV04] about random projections. The latter result offers a simplified hyperplane rounding algorithm for the computation of an O(√logn)-approximation to the Sparsest Cut problem with uniform demands.