Algorithmic Extensions of Cheeger’s Inequality to Higher Eigenvalues and Partitions

Abstract

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 $8\sqrt{1-\lambda_m} \log m$ where λ _m is the m ^th largest eigenvalue of the normalized adjacency matrix. For sparsest small-set, we bound the sparsity by $O(\sqrt{(1-\lambda_{m^2})\log m})$.