Geometry based heuristics for unit disk graphs

TL;DR

This paper introduces simple, provably effective heuristics for classical NP-hard problems on unit disk graphs, using geometric insights to achieve good approximation ratios without requiring geometric representations.

Contribution

It presents new heuristics for key NP-hard problems on unit disk graphs that do not depend on geometric representations, with proven performance guarantees.

Findings

Heuristics for maximum independent set, vertex cover, coloring, and dominating set with performance guarantees.

An online coloring heuristic with a competitive ratio of 6.

Extensions to other intersection graphs and higher-dimensional objects.

Abstract

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.