Distributed $\Delta$-Coloring Plays Hide-and-Seek

TL;DR

This paper establishes new tight lower bounds for distributed symmetry breaking problems, including $ ext{Delta}$-coloring and related variants, using a novel fixed point approach within the round elimination framework.

Contribution

It introduces a new proof technique for lower bounds, extends results to arbdefective colorings, and provides a unified framework for various symmetry breaking problems.

Findings

Deterministic $ ext{Delta}$-coloring on $ ext{Delta}$-regular trees requires $ ext{Omega}( ext{log}_ ext{Delta} n)$ rounds.

Randomized algorithms for the same problem require $ ext{Omega}( ext{log}_ ext{Delta} ext{log} n)$ rounds.

Established tight lower bounds for $(2,eta)$-ruling set and other symmetry breaking problems.

Abstract

We prove several new tight distributed lower bounds for classic symmetry breaking graph problems. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a $\Delta$-coloring on $\Delta$-regular trees requires $\Omega(\log_\Delta n)$ rounds and any randomized algorithm requires $\Omega(\log_\Delta\log n)$ rounds. We prove this result by showing that a natural relaxation of the $\Delta$-coloring problem is a fixed point in the round elimination framework. As a first application, we show that our $\Delta$-coloring lower bound proof directly extends to arbdefective colorings. We exactly characterize which variants of the arbdefective coloring problem are "easy", and which of them instead are "hard". As a second application, which we see as our main contribution, we use the structure of the fixed point as a building block to prove lower bounds as a function of $\Delta$ for a large class of distributed symmetry breaking problems. For example, we obtain a tight lower bound for the fundamental problem of computing a $(2,\beta)$-ruling set. This is an exponential improvement over the best existing lower bound for the problem, which was proven in [FOCS '20]. Our lower bound even applies to a much more general family of problems that allows for almost arbitrary combinations of natural constraints from coloring problems, orientation problems, and independent set problems, and provides a single unified proof for known and new lower bound results for these types of problems. Our lower bounds as a function of $\Delta$ also imply lower bounds as a function of $n$. We obtain, for example, that maximal independent set, on trees, requires $\Omega(\log n / \log \log n)$ rounds for deterministic algorithms, which is tight.