A Reformulation of the Arora-Rao-Vazirani Structure Theorem

A reformulation of the Arora-Rao-Vazirani Structure Theorem Sanjeev Arora∗ James Lee† Sushant Sachdeva‡ October 23, 2018 Abstract In a well-known paper[5], Arora, Rao and Vazirani obtained an O(√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 [5] in terms of the expansion of large sets in geometric graphs on sets of points satisfying triangle inequalities. 1 Introduction Definition 1.1 (Triangle Inequalities). A set of points V is said to satisfy triangle inequalities if for every vi, vj, vk ∈V , the following inequality holds ∥vi −vj∥2 + ∥vj −vk∥2 ≥∥vi −vk∥2 For a set of points V , we define average squared distance to be the expression Ei,j ∥vi −vj∥2 where the expectation is taken over all pairs i, j ∈V . The following geometric theorem was shown in the well-known paper by Arora, Rao and Vazirani[5]. This theorem and its variants underlie subsequent work on improved approximation algorithms for several fundamental problems [1, 8, 7] and metric embeddings [6, 4]. Theorem 1.2 (ARV structure theorem (Theorem 1) [5]; existence of “well-separated sets”). For every c > 0 there exist c′, b > 0 such that the following holds for all n, d: Given n points on unit (d −1)-sphere, v1, . . . , vn ∈Rd that satisfy triangle inequality such that the average squared distance is c, then there exist two sets S, T ⊆{vi}i∈[n] of size at least c′n such that for every vi ∈S, vj ∈T, ∥vi −vj∥2 ≥ b √log n There has been subsequent work on efficient algorithms for Uniform Sparsest Cut and the Balanced separator problem [2, 3, 10]. These results require efficient algorithmic variants of the structure theorem and are based on the notion of expander flows[5]. Our equivalent formulation concerns the expansion of some family of geometric graphs. Now we define this family. ∗Department of Computer Science and Center for Computational Intractability, Princeton University. Supported by NSF Grants CCF-0832797, 0830673, and 0528414. †Computer Science and Engineering, University of Washington. ‡Department of Computer Science and Center for Computational Intractability, Princeton University. Supported by NSF Grants CCF-0832797, 0830673, and 0528414. 1 arXiv:1102.1456v1 [cs.DM] 7 Feb 2011 Definition 1.3 (GV,ϵ). Given a set of points V ⊆Rd, we define GV,ϵ to be the graph on the vertex set V obtained by adding an edge between any two points vi, vj such that ∥vi −vj∥2 ≤ϵ. For any ϵ ≥0, GV,ϵ has a self-loop at each vertex. Thus if Γ(S) denotes the set of neighbors of S, S ⊆Γ(S). Our reformulation will talk about the expansion of large sets in graphs GV,ϵ. We will use the following definition of an expander. This definition is not really standard but has been tailored to improve readability. Definition 1.4. A graph G is said to be an (α, β)-expander if for every set S of size α|V (G)| ≤ |S| ≤ 1 2β|V (G)|, we have |Γ(S)| > β|S| (where Γ(S) denotes the set of neighbors of S). Note that the definition requires a lower bound on the size of the set S. Also note the strict inequality in the requirement for the size of Γ(S). For the graphs that we care about, S ⊆Γ(S) and hence β > 1 for the definition to be non-trivial. Observe that an (α, β)-expander is also an (α′, β′)-expander for all α′ ≥α and β′ ≤β. In this note we prove that the following is an equivalent reformulation of the ARV Structure theorem. Theorem 1.5 (Main). For every c > 0, there exist γ > 0, 1 2 > α > 0 such that the following holds for all n, d, ϵ and β > 1: Given a set V of n points on the unit sphere that satisfy the triangle inequality condition such that their average squared distance is at least c, and GV,ϵ is an (α, β)-node expander, then ϵ ≥ γ k√log n where k =  logβ 1 2α  (or equivalently n ≥exp(γ2/k2ϵ2)). 2 Proof of the main theorem Proof. (1.5 ⇒1.2) Given a set V of n points on the unit sphere that satisfies the conditions of theorem 1.2, we construct the graph GV,ϵ for ϵ = γ 3√log n. Now, using theorem 1.5 with the same c and β2 = 1 2α, there exists a non-trivial α such that GV,ϵ is not an  α, q 1 2α  -expander (since k = 2 and ϵ < γ 2√log n). Thus, there exists a set S such that αn ≤|S| ≤ 1 2βn = p α 2 n such that Γ(S) ≤β|S| ≤1 2n. This means that there is a set T = V \Γ(S) of size at least 1 2n such that there are no edges between S and T. Thus S and T are sets of size at least c′n (for c′ = α) such that there is no edge between them in GV,ϵ. Hence they are b √log n-separated for b = γ 3 (by the definition of GV,ϵ). (1.2 ⇒1.5) Given n points on the unit sphere that satisfy the conditions of theorem 1.5, we use theorem 1.2 with the same c to get two sets S, T of size at least c′n such that they are b √log n separated. Without loss of generality, we shall assume that |S| ≤|T|. Assume t

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