From descriptive to distributed

TL;DR

The paper investigates the interplay between descriptive set theory (measurable/ Borel graph colorings) and distributed computing (LOCAL model) with a focus on vertex and edge colorings. It develops a cross-domain program: (i) import Marks' determinacy and homomorphism-graph techniques to prove lower bounds and transfer results between the infinite and finite worlds, and (ii) extend Vizing-type colorings through multi-step Vizing chains to obtain measurable and distributed colorings. Key contributions include adapting Marks' game method to the LOCAL model, establishing transfer principles via homomorphism graphs, and presenting a deterministic distributed Brooks coloring algorithm for graphs of subexponential growth, along with a comprehensive synthesis of measurable Brooks and Vizing colorings. The work clarifies how measurable combinatorics informs distributed algorithms and vice versa, providing new algorithms, lower bounds, and a rich set of open problems that bridge finite and infinite coloring theory and their complexity landscapes.

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.

Figures (5)

• Figure 1: (Courtesy of Jukka Soumela) The blue dots represent classes of LCL problems (size of the dots suggest the importance of the class), that is, every LCL problem, e.g., vertex coloring with $\Delta$ colors, belongs to one of the dots. Deterministic LOCAL complexity is given by the projection of the dot to the horizontal axis and randomized LOCAL complexity is given by the projection to the diagonal. Light salmon colored areas do not contain any complexity class of LCL problems, in particular, they represent the speed-up results.
• Figure 2: The game $\mathbb{G}(v,i)$
• Figure 3: Strategy stealing
• Figure 4: An example of a Vizing chain $W(x,e)={F_e}^\frown P_{\alpha_e,\beta_e}$.
• Figure 5: An example of a $3$-step Vizing chain. Fans are depicted by black thick edges, while dotted curly edges indicate how the alternating paths continue after they get truncated.

Theorems & Definitions (89)

• Theorem 2.1: Greedy coloring
• Theorem 2.2: Brooks coloring
• Theorem 2.3: Vizing coloring
• Definition 2.4: Augmenting subgraph
• Theorem 2.5: Small augmenting subgraphs for edge colorings, Christiansen Christiansen
• Example 3.1: Color reduction algorithm, Section 3.5 in barenboimelkin_book
• Definition 3.2: Distributed algorithm
• Definition 3.3: LOCAL complexity of graph colorings
• Theorem 3.4: Distributed Greedy coloring, Cole--Vishkin cole86, Goldberg--Plotkin--Shannon goldberg88, Linial linial92LOCAL
• Theorem 3.5: Distributed lower bounds for colorings, brandt_etal2016LLLChangHLPU20