Finding Sparse Cuts via Cheeger Inequalities for Higher Eigenvalues

Cheeger’s fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance of S or the sparsity of the cut (S, S̄)) is bounded as follows: φ(S) def = w(S, S̄) min{w(S), w(S̄)} 6 √ 2λ2, 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. We study three natural generalizations of the sparsest cut in a graph: • a partition of the vertex set into k parts that minimizes 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 k − 1 parts); • a collection of k disjoint subsets S1, . . . , Sk that minimize maxi∈[k] φ(Si); • a subset of size O(1/k) of the graph with minimum expansion. 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 k-partition, we prove that the sparsity is at most 8 √ λk log k where λk is the k th smallest eigenvalue of the normalized Laplacian matrix. For the k sparse cuts problem we prove that there exist ck disjoint subsets S1, . . . , Sck, such that max i φ(Si) 6 C √ λk log k where c, C are suitable absolute constants; this leads to a similar 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) 6 C √ λk log k. The latter two results are the best possible in terms of the eigenvalues up to constant factors. Our results are derived via simple and efficient algorithms, and can themselves be viewed as generalizations of Cheeger’s method. ∗Supported by National Science Foundation awards AF-0915903 and AF-0910584. †Supported by National Science Foundation Career Award and Alfred P. Sloan Fellowship. ‡Supported by National Science Foundation awards DMS-1101447 and AF-0910584.