Min-max graph partitioning and small set expansion

Nikhil Bansal, Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph Seffi Naor, Roy Schwartz

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( v log n log k) approximation algorithm. This improves over an O(log2 n) approximation for the second version due to Svitkina and Tardos [Min-max multiway cut, in APPROX-RANDOM, 2004, Springer, Berlin, 2004], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an O(1) approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the small-set expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ? V of size |S| ≤pn with minimum edge expansion. We give an O(log n log (1/p)) bicriteria approximation algorithm for small-set expansion in general graphs, and an improved factor of O(1) for graphs that exclude any fixed minor.

Original languageEnglish (US)
Pages (from-to)872-904
Number of pages33
JournalSIAM Journal on Computing
Issue number2
StatePublished - 2014


  • Balanced cut
  • Expansion
  • Sparse cut
  • Spreading metrics

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics


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