Hard Metrics from Cayley Graphs of Abelian Groups

Ilan Newman,Yuri Rabinovich

doi:10.4086/toc.2009.v005a006

Abstract

Hard metrics are the class of extremal metrics with respect to embedding into Euclidean spaces; they incur $\Omega(\log n)$ multiplicative distortion, which is as large as it can possibly get for any metric space of size $n$. Besides being very interesting objects akin to expanders and good error-correcting codes, and having a rich structure, such metrics are important for obtaining lower bounds in combinatorial optimization, e.g., on the value of MinCut/MaxFlow ratio for multicommodity flows. For more than a decade, a single family of hard metrics was known (Linial, London, Rabinovich (Combinatorica 1995) and Aumann, Rabani (SICOMP 1998)). Recently, a different family was found by Khot and Naor (FOCS 2005). In this paper we present a general method of constructing hard metrics. Our results extend to embeddings into negative type metric spaces and into $\ell_1.$

Highlights

In this paper we present a general method of constructing hard metrics
Our results extend to embeddings into negative type metric spaces and into 1
A famous theorem of Bourgain [4] states that every finite metric space V = (X, d) of size n = |X| can be embedded into a Euclidean space with multiplicative distortion at most O(log n)

Summary

Introduction

We call a metric space V hard with respect to 2 if any embedding of V into a Euclidean space (of any dimension) has a multiplicative distortion Ω(log n). It is always possible to choose a set A of generators for H, so that the shortest-path metric of the corresponding Cayley graph G(H, A) is hard with respect to 2, 1 and NEG. In the special case of Zn2, good sets of generators are closely related to error-correcting codes of constant rate and linear distance Our construction is both simple to describe and easy to analyze. Note: in what follows we restrict the discussion to Euclidean spaces, the same method can be used to show the hardness of the metrics that we construct with respect to the much richer class NEG of “negative type metrics” and to 1

Definitions

Abelian Groups

When the Group is Zn2

Additional Remarks