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, that has appeared in the context of understanding the unique games conjecture, 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 Laplacian. In particular, for the sparsest m-partition, we prove that the sparsity is at most 8 √ 1 − λm log m where λm is the m th largest eigenvalue of the normalized adjacency matrix. For sparsest small-set, we bound the sparsity by O( (1 − λ m 2 ) log m). Our results are algorithmic, with the first using a recursive spectral decomposition and the second using a convex relaxation.
where vol(S) is defined as the sum of the degrees of the vertices in S. REF gave an algorithm to find a k-partition which cuts at most O √ λ k log k fraction of the edges.
Algorithmic Extensions of Cheeger’s Inequality to Higher Eigenvalues and Partitions
APPROX-RANDOM
Mathematics Computer Science