Local Global Tradeoffs in Metric Embeddings

Suppose that every k points in a metric space X are D-distortion embeddable into lscr 1. We give upper and lower bounds on the distortion required to embed the entire space X into lscr 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 lscr 1 with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.

[1]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[2]  Venkatesan Guruswami,et al.  Maximum-likelihood decoding of Reed-Solomon codes is NP-hard , 2004, SODA.

[3]  A. Naor,et al.  Euclidean quotients of finite metric spaces , 2004, math/0406349.

[4]  Venkatesan Guruswami.  8 List Decoding of Concatenated Codes , 2004 .

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

[6]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[7]  Madhur Tulsiani,et al.  Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut , 2007, STOC '07.

[8]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[9]  Madhu Sudan,et al.  Decoding of Reed Solomon Codes beyond the Error-Correction Bound , 1997, J. Complex..

[10]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[11]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[12]  Santosh S. Vempala,et al.  Local versus global properties of metric spaces , 2006, SODA '06.

[13]  K. Menger.  Untersuchungen über allgemeine Metrik , 1928 .

[14]  Eyal Kushilevitz,et al.  Learning decision trees using the Fourier spectrum , 1991, STOC '91.

[15]  Qi Cheng,et al.  On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract) , 2004, FOCS.

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

[17]  Venkatesan Guruswami,et al.  List decoding of error correcting codes , 2001 .

[18]  N. Tomczak-Jaegermann.  Banach-Mazur distances and finite-dimensional operator ideals , 1989 .

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

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

[21]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[22]  Wenceslas Fernandez de la Vega,et al.  Linear programming relaxations of maxcut , 2007, SODA '07.

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

[24]  Rocco A. Servedio,et al.  Agnostically Learning Halfspaces , 2005, FOCS.

[25]  Ronitt Rubinfeld,et al.  Learning Polynomials with Queries: The Highly Noisy Case , 2000, SIAM J. Discret. Math..

[26]  Béla Bollobás,et al.  Proving Integrality Gaps without Knowing the Linear Program , 2006, Theory Comput..

[27]  Vitaly Feldman.  Optimal hardness results for maximizing agreements with monomials , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[28]  J. Matousek,et al.  On embedding expanders into ℓp spaces , 1997 .

[29]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[30]  J. Bourgain.  The metrical interpretation of superreflexivity in banach spaces , 1986 .

[31]  Noam Nisan,et al.  Constant depth circuits, Fourier transform, and learnability , 1993, JACM.

[32]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

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

[34]  Amnon Ta-Shma,et al.  Extractors from Reed-Muller codes , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

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

[36]  Michael Dinitz,et al.  Spanners with Slack , 2006, ESA.

[37]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[38]  E. Helly.  Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. , 1923 .

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

[40]  Madhu Sudan,et al.  Improved Low-Degree Testing and its Applications , 2003, Comb..

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

[42]  James R. Lee,et al.  Metric Structures in L1: Dimension, Snowflakes, and Average Distortion , 2004, LATIN.

[43]  Ittai Abraham,et al.  Reconstructing approximate tree metrics , 2007, PODC '07.

[44]  Vitaly Feldman,et al.  New Results for Learning Noisy Parities and Halfspaces , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[45]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[46]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[47]  Moses Charikar,et al.  Integrality gaps for Sherali-Adams relaxations , 2009, STOC '09.

[48]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.