Approximation Algorithms for Semi-random Graph Partitioning Problems

TL;DR

This paper introduces a flexible semi-random graph partitioning model and develops approximation algorithms that work across various problems and models, including new planted algebraic expanders.

Contribution

It proposes a new semi-random model capturing real-world properties and provides general approximation algorithms applicable to multiple graph partitioning problems.

Findings

Constant factor bi-criteria approximations for several problems

Algorithms work in wider parameter ranges than previous models

Almost recovery of optimal solutions under expansion conditions

Abstract

In this paper, we propose and study a new semi-random model for graph partitioning problems. We believe that it captures many properties of real--world instances. The model is more flexible than the semi-random model of Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and Sipser. We develop a general framework for solving semi-random instances and apply it to several problems of interest. We present constant factor bi-criteria approximation algorithms for semi-random instances of the Balanced Cut, Multicut, Min Uncut, Sparsest Cut and Small Set Expansion problems. We also show how to almost recover the optimal solution if the instance satisfies an additional expanding condition. Our algorithms work in a wider range of parameters than most algorithms for previously studied random and semi-random models. Additionally, we study a new planted algebraic expander model and develop constant factor bi-criteria approximation algorithms for graph partitioning problems in this model.