An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Lectures on Discrete Geometry
Cuts, Trees and -Embeddings of Graphs
FOCS '99 Proceedings of the 40th Annual 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
0(\sqrt {\log n)} Approximation to SPARSEST CUT in Õ(n2) Time
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
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
On distance scales, embeddings, and efficient relaxations of the cut cone
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Improved lower bounds for embeddings into L1
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Integrality gaps for sparsest cut and minimum linear arrangement problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Metric structures in L1: dimension, snowflakes, and average distortion
European Journal of Combinatorics
Lp metrics on the Heisenberg group and the Goemans-Linial conjecture
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
How to Play Unique Games Using Embeddings
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Volume Distortion for Subsets of Euclidean Spaces
Discrete & Computational Geometry
On the geometry of graphs with a forbidden minor
Proceedings of the forty-first annual ACM symposium on Theory of computing
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
A $(\log n)^{\Omega(1)}$ Integrality Gap for the Sparsest Cut SDP
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Near-optimal distortion bounds for embedding doubling spaces into L1
Proceedings of the forty-third annual ACM symposium on Theory of computing
On the 2-sum embedding conjecture
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We show that there exists a metric space (X,d) such that (X,√d) admits a bilipschitz embedding into L2, but (X,d) does not admit an equivalent metric of negative type. In fact, we exhibit a strong quantitative bound: There are n-point subsets Yn ⊆ X such that mapping (Yn, d) to a metric of negative type requires distortion ~Ω(log n)1/4. In a formal sense, this is the first lower bound specifically against bilipschitz embeddings into negative-type metrics, and therefore unlike other lower bounds, ours cannot be derived from a 1-dimensional Poincare inequality. This answers an open question about the strength of strong vs. weak triangle inequalities in a number of semi-definite programs. Our construction sheds light on the power of various notions of "dual flows" that arise in algorithms for approximating the Sparsest Cut problem. It also has other interesting implications for bilipschitz embeddings of finite metric spaces.