Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut

In this paper, we study the metrics of <i>negative type</i>, which are metrics (<i>V</i>, d) such that √d is an Euclidean metric; these metrics are thus also known as "<i>l</i><inf>2</inf>-squared" metrics.We show how to embed <i>n</i>-point negative-type metrics into Euclidean space <i>l</i><inf>2</inf> with distortion <i>D</i> = <i>O</i>(log<sup>3/4</sup> <i>n</i>). This embedding result, in turn, implies an <i>O</i>(log<sup>3/4</sup> <i>k</i>)-approximation algorithm for the Sparsest Cut problem with non-uniform demands. Another corollary we obtain is that <i>n</i>-point subsets of <i>l</i><inf>1</inf> embed into <i>l</i><inf>2</inf> with distortion <i>O</i>(log<sup>3/4</sup> <i>n</i>).

[1]  P. Enflo.  On the nonexistence of uniform homeomorphisms betweenLp-spaces , 1970 .

[2]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[3]  Alexander V. Karzanov,et al.  Metrics and undirected cuts , 1985, Math. Program..

[4]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[5]  Farhad Shahrokhi,et al.  The maximum concurrent flow problem , 1990, JACM.

[6]  Manfred K. Warmuth,et al.  The Weighted Majority Algorithm , 1994, Inf. Comput..

[7]  Kenneth L. Clarkson,et al.  Las Vegas algorithms for linear and integer programming when the dimension is small , 1995, JACM.

[8]  Nathan Linial,et al.  The geometry of graphs and some of its algorithmic applications , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[9]  David B. Shmoys,et al.  Cut problems and their application to divide-and-conquer , 1996 .

[10]  Emo Welzl.  Suchen und Konstruieren durch Verdoppeln , 1996 .

[11]  J. Matousek,et al.  On the distortion required for embedding finite metric spaces into normed spaces , 1996 .

[12]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[13]  Michel X. Goemans,et al.  Semidefinite programming in combinatorial optimization , 1997, Math. Program..

[14]  M. Goemans.  Semidefinite programming and combinatorial optimization , 1998 .

[15]  Yuval Rabani,et al.  An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm , 1998, SIAM J. Comput..

[16]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[17]  Satish Rao,et al.  Small distortion and volume preserving embeddings for planar and Euclidean metrics , 1999, SCG '99.

[18]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[19]  J. Matousek,et al.  On embedding trees into uniformly convex Banach spaces , 1999 .

[20]  J. M. Sek.  On embedding trees into uniformly convex Banach spaces , 1999 .

[21]  Piotr Indyk,et al.  Algorithmic applications of low-distortion geometric embeddings , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[22]  Yuval Rabani,et al.  Approximation algorithms for the 0-extension problem , 2001, SODA '01.

[23]  Nathan Linial.  Finite metric spaces: combinatorics, geometry and algorithms , 2002, SCG '02.

[24]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[25]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[26]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

[27]  I. Bárány.  LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) , 2003 .

[28]  Robert Krauthgamer,et al.  Bounded geometries, fractals, and low-distortion embeddings , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[29]  Imre Brny.  LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) By JI MATOUEK: 481 pp., 31.50 (US$39.95), ISBN 0-387-95374-4 (Springer, New York, 2002). , 2003 .

[30]  Yuri Rabinovich.  On average distortion of embedding metrics into the line and into L1 , 2003, STOC '03.

[31]  Anupam Gupta,et al.  Embedding k-outerplanar graphs into ℓ1 , 2003, SODA '03.

[32]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[33]  Kunal Talwar,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2004, J. Comput. Syst. Sci..

[34]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[35]  Robert Krauthgamer,et al.  Measured descent: a new embedding method for finite metrics , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[36]  Robert Krauthgamer,et al.  Measured Descent: A New Embedding Method for Finite Metrics , 2004, FOCS.

[37]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[38]  James R. Lee,et al.  On distance scales, embeddings, and efficient relaxations of the cut cone , 2005, SODA '05.

[39]  Nisheeth K. Vishnoi,et al.  The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into l1 , 2005, FOCS.

[40]  James R. Lee,et al.  Extending Lipschitz functions via random metric partitions , 2005 .

[41]  Yuval Rabani,et al.  ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[42]  James R. Lee,et al.  Euclidean distortion and the sparsest cut , 2005, STOC '05.

[43]  Y. Rabani,et al.  Improved lower bounds for embeddings into L 1 , 2006, SODA 2006.