Unbalanced Graph Partitioning

Abstract

We investigate the unbalanced cut problems. A cut (A, B) is called unbalanced if the size of its smaller side is at most k (called k-size) or exactly k (called Ek-size), where k is an input parameter. An s-t cut (A, B) is called unbalanced if its s-side is of k-size or Ek-size. We consider three types of unbalanced cut problems, in which the quality of a cut is measured with respect to the capacity, the sparsity, and the conductance, respectively.

We show that even if the input graph is restricted to be a tree, the Ek-Sparsest Cut problem (to find an Ek-size cut with the minimum sparsity) is still NP-hard. We give a bicriteria approximation algorithm for the k-Sparsest Cut problem (to find a k-size cut with the minimum sparsity), which outputs a cut whose sparsity is at most O(logn) times the optimum and whose smaller side has size at most O(logn)k. As a consequence, this leads to a (O(logn), O(logn))-approximation algorithm for the Min k-Conductance problem (to find a k-size cut with the minimum conductance). We also prove that the Min k-Size s-t Cut problem is NP-hard and give an O(logn)-approximation algorithm for it.