Combinatorica
An improved spectral graph partitioning algorithm for mapping parallel computations
SIAM Journal on Scientific Computing
An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Spectral partitioning works: planar graphs and finite element meshes
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
On clusterings: Good, bad and spectral
Journal of the ACM (JACM)
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Spectral Partitioning, Eigenvalue Bounds, and Circle Packings for Graphs of Bounded Genus
SIAM Journal on Computing
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Eigenvalue bounds, spectral partitioning, and metrical deformations via flows
Journal of the ACM (JACM)
Approximations for the isoperimetric and spectral profile of graphs and related parameters
Proceedings of the forty-second ACM symposium on Theory of computing
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Subexponential Algorithms for Unique Games and Related Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Algorithmic extensions of cheeger's inequality to higher eigenvalues and partitions
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Min-max Graph Partitioning and Small Set Expansion
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Multi-way spectral partitioning and higher-order cheeger inequalities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Multi-way spectral partitioning and higher-order cheeger inequalities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance) is bounded as follows: [ φ(S) def= (w(S,bar{S}))/(min set(w(S), w(bar(S)))) ≤ √(2 λ2) ] where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer k ∈ [n], there exist ck disjoint subsets S1, ..., Sck, such that [ maxi φ(Si) ≤ C √(λk log k) ] where λk is the kth smallest eigenvalue of the normalized Laplacian and c0 are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any k, there is a subset S whose weight is at most a O(1/k) fraction of the total weight and φ(S) ≤ C √(λk log k). Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding k subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right.