Unbalanced Graph Partitioning

  • Authors:
  • Angsheng Li;Peng Zhang

  • Affiliations:
  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China 100190;School of Computer Science and Technology, Shandong University, Jinan, China 250101

  • Venue:
  • Theory of Computing Systems
  • Year:
  • 2013

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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. We consider two closely related unbalanced cut problems, in which the quality of a cut is measured with respect to 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))-bicriteria approximation algorithm for the Min k-Conductance problem (to find a k-size cut with the minimum conductance).