Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

TL;DR

This paper investigates intersection graphs of low-density objects in Euclidean space, establishing small separators and providing efficient approximation algorithms for key problems, while also characterizing their computational hardness.

Contribution

It introduces approximation algorithms for low-density intersection graphs and characterizes their intractability based on density, extending understanding of these graph classes.

Findings

Low-density graphs have small separators.

Efficient $(1+ ext{epsilon})$-approximation algorithms are developed.

Some problems are proven hard to approximate within certain bounds.

Abstract

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density.