## Abstract

We consider graph coloring and related problems in the distributed message-passing model. Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop-neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative Δ+ 1-coloring algorithm requires ω (Δlog Δ+ log ∗ n) rounds, unless there exists "a very special type of coloring that can be very efficiently reduced"[44]. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms and to explore other approaches to the coloring problem [2, 3, 19, 32]. The latter gave rise to faster algorithms, but their heavy machinery that is of non-locally-iterative nature made them far less suitable to various settings. In this article, we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative Δ+ 1-coloring algorithm with running time O(Δ+ log ∗ n), i.e., below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing, and bandwidth-restricted settings. This includes the following results:We obtain self-stabilizing distributed algorithms for Δ+ 1-vertex-coloring, (2Δ- 1)-edge-coloring, maximal independent set, and maximal matching with O(Δ+ log ∗ n) time. This significantly improves previously known results that have O(n) or larger running times [23].We devise a (2Δ- 1)-edge-coloring algorithm in the CONGEST model with O(Δ+ log ∗ n) time and O(δ)-edge-coloring in the Bit-Round model with O(Δ+ log n) time. The factors of log ∗ n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously known algorithms had superlinear dependency on Δfor (2Δ- 1)-edge-coloring in these models.We obtain an arbdefective coloring algorithm with running time O( Δ+ log ∗ n). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it to compute a proper (1 + ϵ)δ-coloring within O( Δ+ log ∗ n) time and Δ+ 1-coloring within O( Δlog Δlog ∗ Δ+ log ∗ n) time. This improves the recent state-of-the-art bounds of Barenboim from PODC'15 [2] and Fraigniaud et al. from FOCS'16 [19] by polylogarithmic factors.Our algorithms are applicable to the SET-LOCAL model [25] (also known as the weak LOCAL model). In this model a relatively strong lower bound of ω (Δ1/3) is known for Δ+ 1-coloring. However, most of the coloring algorithms do not work in this model. (In Reference [25] only Linial's O(Δ2)-time algorithm and Kuhn-Wattenhofer O(Δlog δ)-time algorithms are shown to work in it.) We obtain the first linear-in-ΔΔ+ 1-coloring algorithms that work also in this model.

Original language | English |
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Article number | 5 |

Journal | Journal of the ACM |

Volume | 69 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2022 |

## Keywords

- Symmetry-breaking
- congest networks
- self-stabilization

## ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence