Partitioning into Expanders
Shayan Oveis Gharan, Luca Trevisan

TL;DR
This paper introduces a spectral algorithm for partitioning graphs into expanders with high conductance, providing robust theoretical guarantees and practical clustering applications without relying on higher order eigenfunctions.
Contribution
The paper presents a new polynomial-time spectral algorithm for graph partitioning into expanders, with robustness and improved conductance guarantees, avoiding the use of higher order eigenfunctions.
Findings
Algorithm finds partitions with low conductance sets and high conductance subgraphs.
Performance improves with a spectral gap between lambda_k and lambda_{k+1}.
Algorithm is simple, local, and effective for clustering applications.
Abstract
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\leq \ell\leq k-1, V can be {\em partitioned} into l sets P_1,\ldots,P_l such that each P_i is a low-conductance set in G and induces a high conductance induced subgraph. In particular, \phi(P_i)=O(l^3\sqrt{\lambda_l}) and \phi(G[P_i]) >= \lambda_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…
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Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Matrix Theory and Algorithms
