Min-Max Graph Partitioning and Small Set Expansion

TL;DR

This paper introduces improved approximation algorithms for min-max graph partitioning problems, including a new approach to the Small-Set Expansion problem, achieving better bounds especially for minor-free graphs.

Contribution

It presents a novel approximation algorithm for a generalized min-max graph partitioning problem and an improved method for Small-Set Expansion, with better bounds for minor-free graphs.

Findings

Achieves an $O(\sqrt{\log n\log k})$-approximation for generalized min-max partitioning.

Provides an $O(\sqrt{\log n\log (1/ ho)})$ bicriteria approximation for Small-Set Expansion.

Offers an $O(1)$ approximation for graphs excluding any fixed minor.

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.