Partitioning into Expanders

Let G = (V,E) be an undirected graph, λk be the k th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that λk > 0 if and only if G has at most k − 1 connected components. We prove a robust version of this fact. If λk > 0, then for some 1 ≤ l ≤ k − 1, V can be partitioned into l sets P1, . . . , Pl such that each Pi is a low-conductance set in G and induces a high conductance induced subgraph. In particular, φ(Pi) . l 3 √ λl and φ(G[Pi]) & λk/k . We make our results algorithmic by designing a simple polynomial time spectral algorithm to find such partitioning of G with a quadratic loss in the inside conductance of Pi’s. Unlike the recent results on higher order Cheeger’s inequality [LOT12, LRTV12], our algorithmic results do not use higher order eigenfunctions of G. In addition, if there is a sufficiently large gap between λk and λk+1, more precisely, if λk+1 & poly(k)λ 1/4 k then our algorithm finds a k partitioning of V into sets P1, . . . , Pk such that the induced subgraph G[Pi] has a significantly larger conductance than the conductance of Pi in G. Such a partitioning may represent the best k clustering of G. Our algorithm is a simple local search that only uses the Spectral Partitioning algorithm as a subroutine. We expect to see further applications of this simple algorithm in clustering applications. Let ρ(k) = mindisjoint A1,...,Ak max1≤i≤k φ(Ai) be the order k conductance constant of G, in words, ρ(k) is the smallest value of the maximum conductance of any k disjoint subsets of V . Our main technical lemma shows that if (1 + ǫ)ρ(k) < ρ(k + 1), then V can be partitioned into k sets P1, . . . , Pk such that for each 1 ≤ i ≤ k, φ(G[Pi]) & ǫ · ρ(k + 1)/k and φ(Pi) ≤ k · ρ(k). This significantly improves a recent result of Tanaka [Tan12] who assumed an exponential (in k) gap between ρ(k) and ρ(k + 1). Computer Science Division, U.C. Berkeley. This material is supported by a Stanford graduate fellowship and a Miller fellowship. Email:[email protected] . Department of Computer Science, Stanford University. This material is based upon work supported by the National Science Foundation under grants No. CCF 1017403 and CCF 1216642. Email:[email protected]. Figure 1: In this example although both sets in the 2-partitioning are of small conductance, in a natural clustering the red vertex (middle vertex) will be merged with the left cluster