Distributed coloring and the local structure of unit-disk graphs

TL;DR

This paper presents new distributed algorithms for coloring unit-disk graphs with bounds related to their clique number, improving efficiency and providing structural insights, with potential for near-optimal coloring in distributed settings.

Contribution

It introduces a constant-time distributed coloring algorithm with at most 4 times the clique number colors in the location-aware setting and a logarithmic-round algorithm with about 5.68 times the clique number colors without location knowledge, along with structural conjectures.

Findings

Constant-time algorithm colors with 4ω(G) colors in location-aware setting.

O(log* n) rounds algorithm colors with 5.68ω(G)+1 colors without location info.

Study of local structure of unit-disk graphs and conjecture on average degree bound.

Abstract

Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In this context it is important to bound not only the complexity of the coloring algorithms, but also the number of colors used. In this paper, we consider two natural distributed settings. In the location-aware setting (when nodes know their coordinates in the plane), we give a constant time distributed algorithm coloring any unit-disk graph $G$ with at most $4\omega(G)$ colors, where $\omega(G)$ is the clique number of $G$. This improves upon a classical 3-approximation algorithm for this problem, for all unit-disk graphs whose chromatic number significantly exceeds their clique number. When nodes do not know their coordinates in the plane, we give a distributed algorithm in the LOCAL model that colors every unit-disk graph $G$ with at most $5.68\omega(G)+1$ colors in $O(\log^* n)$ rounds. This algorithm is based on a study of the local structure of unit-disk graphs, which is of independent interest. We conjecture that every unit-disk graph $G$ has average degree at most $4\omega(G)$, which would imply the existence of a $O(\log n)$ round algorithm coloring any unit-disk graph $G$ with (approximately) $4\omega(G)$ colors in the LOCAL model. We provide partial results towards this conjecture using Fourier-analytical tools.