Towards Fully Automatic Distributed Lower Bounds

TL;DR

This paper introduces an automatic method to derive lower bounds for distributed graph problems using round elimination, demonstrated on defective coloring, simplifying the process of proving complexity bounds.

Contribution

The paper presents a fully automatic approach for deriving lower bounds in distributed algorithms via round elimination, including a new method for computing easier problems.

Findings

Automated lower bounds for deterministic and randomized algorithms.

Lower bounds established for defective coloring problems.

Simplified procedure for applying round elimination.

Abstract

In the past few years, a successful line of research has lead to lower bounds for several fundamental local graph problems in the distributed setting. These results were obtained via a technique called round elimination. On a high level, the round elimination technique can be seen as a recursive application of a function that takes as input a problem $\Pi$ and outputs a problem $\Pi'$ that is one round easier than $\Pi$. Applying this function recursively to concrete problems of interest can be highly nontrivial, which is one of the reasons that has made the technique difficult to approach. The contribution of our paper is threefold. Firstly, we develop a new and fully automatic method for finding lower bounds of $\Omega(\log_\Delta n)$ and $\Omega(\log_\Delta \log n)$ rounds for deterministic and randomized algorithms, respectively, via round elimination. Secondly, we show that this automatic method is indeed useful, by obtaining lower bounds for defective coloring problems. We show that the problem of coloring the nodes of a graph with $3$ colors and defect at most $(\Delta - 3)/2$ requires $\Omega(\log_\Delta n)$ rounds for deterministic algorithms and $\Omega(\log_\Delta \log n)$ rounds for randomized ones. We note that lower bounds for coloring problems are notoriously challenging to obtain, both in general, and via the round elimination technique. Both the first and (indirectly) the second contribution build on our third contribution -- a new and conceptually simple way to compute the one-round easier problem $\Pi'$ in the round elimination framework. This new procedure provides a clear and easy recipe for applying round elimination, thereby making a substantial step towards the greater goal of having a fully automatic procedure for obtaining lower bounds in the distributed setting.