Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics
DISC '08 Proceedings of the 22nd international symposium on Distributed Computing
On low dimensional local embeddings
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Integrality gaps for Sherali-Adams relaxations
Proceedings of the forty-first annual ACM symposium on Theory of computing
Vertex cover resists SDPs tightened by local hypermetric inequalities
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
On linear and semidefinite programming relaxations for hypergraph matching
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
| Hi-index | 0.01 |
Suppose that every k points in a metric space X are D-distortion embeddable into \ell _1. We give upper and lower bounds on the distortion required to embed the entire space X into \ell _1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into \ell _1 with distortion {\rm O}(D \times \log (\left| X \right|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For D = 1 + \delta we give a lower bound of \Omega (\log (\left| X \right|/k)/\log (1/\delta )); and for D = 1, we give a lower bound of \Omega (\log \left| X \right|/(\log k + \log \log \left| X \right|)). Our bounds significantly improve on the results of Arora, Lovész, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.