Combinatorica
Finding a large hidden clique in a random graph
proceedings of the eighth international conference on Random structures and algorithms
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximate clustering without the approximation
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Eigenvalue bounds, spectral partitioning, and metrical deformations via flows
Journal of the ACM (JACM)
Subexponential Algorithms for Unique Games and Related Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Stability Yields a PTAS for k-Median and k-Means Clustering
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ1 ≤ λ2 ≤ ... ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, [φ(G) = O(k) l2/√lk,] and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any $k$. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if lk is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to spectral algorithms for other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.