From descriptive to distributed

TL;DR

This survey explores how descriptive set theory enhances understanding and development of distributed algorithms for graph coloring problems, including new techniques and open challenges.

Contribution

It introduces novel applications of descriptive set theory to distributed computing, including the adaptation of Marks' games and Vizing's chain techniques for graph coloring.

Findings

Development of the first non-trivial distributed algorithms for Vizing colorings.

Adaptation of Marks' games method to the LOCAL model.

An efficient deterministic algorithm for Brooks coloring on subexponential graphs.

Abstract

In the past couple of years a rich connection has been found between the fields of descriptive set theory and distributed computing. Frequently, and less surprisingly, finitary algorithms can be adopted to the infinite setting, resulting in theorems about infinite, definable graphs. In this survey, we take a different perspective and illustrate how results and ideas from descriptive set theory provide new insights and techniques to the theory of distributed computing. We focus on the two classical topics from graph theory, vertex and edge colorings. After summarizing the up-to-date results from both areas, we discuss the adaptation of Marks' games method to the LOCAL model of distributed computing and the development of the multi-step Vizing's chain technique, which led to the construction of the first non-trivial distributed algorithms for Vizing colorings. We provide a list of related open problems to complement our discussion. Finally, we describe an efficient deterministic distributed algorithm for Brooks coloring on graphs of subexponential growth.