Partitioning into Expanders
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Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that lambda_k > 0 if and only if G has at most k-1 connected components. We prove a robust version of this fact. If lambda_k>0, then for some 1≤ℓ≤ k-1, V can be partitioned into l sets P_1,...,P_l such that each P_i is a low-conductance set in G and induces a high conductance induced subgraph. In particular, ϕ(P_i)=O(l^3√(λ_l)) and ϕ(G[P_i]) >= λ_k/k^2). 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 P_i's. Unlike the recent results on higher order Cheeger's inequality [LOT12,LRTV12], our algorithmic results do not use higher order eigenfunctions of G. If there is a sufficiently large gap between lambda_k and lambda_k+1, more precisely, if λ_k+1 >= (k) lambda_k^1/4 then our algorithm finds a k partitioning of V into sets P_1,...,P_k such that the induced subgraph G[P_i] has a significantly larger conductance than the conductance of P_i 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.
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