A Reformulation of the Arora-Rao-Vazirani Structure Theorem

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.

💡 Research Summary

The paper revisits the celebrated Arora‑Rao‑Vazirani (ARV) Structure Theorem, which lies at the heart of the (O(\sqrt{\log n})) approximation algorithms for Balanced Separator and Uniform Sparsest Cut. The original theorem is a geometric statement: given (n) points on the unit ((d-1))-sphere that satisfy the (\ell_2^2) triangle inequality and have average squared distance at least a constant (c), there exist two subsets (S,T) each of size (\Omega(n)) such that every pair ((u\in S, v\in T)) is separated by distance at least (b\sqrt{\log n}). This “well‑separated” property is the engine behind many subsequent algorithmic results and metric‑embedding works.

The authors propose an equivalent formulation that replaces the distance‑based separation with an expansion property of a natural geometric graph built from the point set. For a parameter (\varepsilon\ge 0) they define the graph (G_{V,\varepsilon}) on vertex set (V) by connecting any two points whose Euclidean distance is at most (\varepsilon) (self‑loops are added so that every vertex belongs to its own neighbourhood). They then introduce a non‑standard but convenient notion of an ((\alpha,\beta))-expander: a graph (G) is an ((\alpha,\beta))-expander if for every vertex subset (S) with (\alpha|V|\le |S|\le \frac12\beta|V|) we have (|\Gamma(S)|>\beta|S|), where (\Gamma(S)) denotes the neighbourhood of (S). The requirement (\beta>1) forces genuine expansion (since (\Gamma(S)\supseteq S) for the graphs under consideration).

Theorem 1.5 (Main Reformulation).
For every constant (c>0) there exist constants (\gamma>0) and (0<\alpha<\frac12) such that the following holds for all (n,d,\varepsilon) and any (\beta>1): if a set (V) of (n) points on the unit sphere satisfies the triangle‑inequality condition and has average squared distance at least (c), and if the graph (G_{V,\varepsilon}) is an ((\alpha,\beta))-expander, then
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