A Reformulation of the Arora-Rao-Vazirani Structure Theorem

TL;DR

This paper provides a new formulation of the Arora-Rao-Vazirani Structure Theorem, linking it to expansion properties of large sets in geometric graphs with triangle inequality constraints, enhancing understanding of approximation algorithms.

Contribution

It offers an equivalent reformulation of the ARV Structure Theorem focusing on expansion in geometric graphs, clarifying the theorem's geometric and combinatorial aspects.

Findings

Equivalent formulation of the ARV theorem in terms of expansion properties

Deeper understanding of the geometric structure underlying approximation algorithms

Potential implications for improved algorithms in graph partitioning

Abstract

In a well-known paper[ARV], Arora, Rao and Vazirani obtained an O(sqrt(log n)) approximation to the Balanced Separator problem and Uniform Sparsest Cut. At the heart of their result is a geometric statement about sets of points that satisfy triangle inequalities, which also underlies subsequent work on approximation algorithms and geometric embeddings. In this note, we give an equivalent formulation of the Structure theorem in [ARV] in terms of the expansion of large sets in geometric graphs on sets of points satisfying triangle inequalities.