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arXiv · 1204.4688

Markov chain methods for small-set expansion

Ryan O'Donnell·David Witmer
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Abstract

Consider a finite irreducible Markov chain with invariant distribution $π$. We use the inner product induced by $π$ and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection between the number of small eigenvalues of a graph's Laplacian and the expansion of small sets in that graph. We give a simplified proof which generalizes to the nonregular, directed case. This result implies an approximation algorithm for an "analytic" version of the Small-Set Expansion Problem, which, in turn, immediately gives an approximation algorithm for Small-Set Expansion. We also give a simpler proof of a lower bound on the probability that a random walk stays within a set; this result was used in some recent works on finding small sparse cuts.

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Ryan O'Donnell, David Witmer. 2013-11-04. Markov chain methods for small-set expansion. https://arxiv.org/abs/1204.4688

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