Combinatorica
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
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
Graph partitioning using single commodity flows
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
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
Multi-way spectral partitioning and higher-order cheeger inequalities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Many sparse cuts via higher eigenvalues
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Clustering and outlier detection using isoperimetric number of trees
Pattern Recognition
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We consider two generalizations of the problem of finding a sparsest cut in a graph. The first is to find a partition of the vertex set into m parts so as to minimize the sparsity of the partition (defined as the ratio of the weight of edges between parts to the total weight of edges incident to the smallest m - 1 parts). The second is to find a subset of minimum sparsity that contains at most a 1/m fraction of the vertices. Our main results are extensions of Cheeger's classical inequality to these problems via higher eigenvalues of the graph. In particular, for the sparsest m-partition, we prove that the sparsity is at most 8√1-λm log m where λm is the mth largest eigenvalue of the normalized adjacency matrix. For sparsest small-set, we bound the sparsity by O(√(1-λm2)log m).