# Min-max graph partitioning and small set expansion

N. Bansal, U. Feige, R. Krauthgamer, K. Makarychev, V. Magarajan, J. Naor, R. Schwartz

35 Citations (Scopus)

## Abstract

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 $O(\sqrt{\log n\log k})$-approximation algorithm. This improves over an $O(\log^2 n)$ approximation for the second version, and roughly $O(k\log n)$ approximation for the first version that follows from other previous work. We also give an improved 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 non-empty set $S\subseteq V$ of size $|S| \leq \rho n$ with minimum edge-expansion. We give an $O(\sqrt{\log{n}\log{(1/\rho)}})$ bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.
Original language English s.n. 29 Published - 2011

### Publication series

Name arXiv.org [cs.DS] 1110.4319

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