Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

TL;DR

This paper explores the deep connections between distributed algorithms, finitary factors, and descriptive combinatorics on trees, establishing new lower bounds, characterizations, and separations among complexity classes using innovative techniques.

Contribution

It extends descriptive combinatorics methods to distributed computing, proving new lower bounds, characterizations, and class separations in the context of local problems on trees.

Findings

Deterministic ( log n) lower bounds for homomorphism problems.

Equivalence between Baire measurable colorings and \u00f3( log n) local algorithms.

Incomparability of sublogarithmic randomized algorithms and Borel solutions.

Abstract

We study connections between distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics in the context of regular trees. We extend the Borel determinacy technique of Marks coming from descriptive combinatorics and adapt it to the area of distributed computing. Using this technique, we prove deterministic distributed $\Omega(\log n)$-round lower bounds for problems from a natural class of homomorphism problems. Interestingly, these lower bounds seem beyond the current reach of the powerful round elimination technique responsible for all substantial locality lower bounds of the last years. Our key technical ingredient is a novel ID graph technique that we expect to be of independent interest. We prove that a local problem admits a Baire measurable coloring if and only if it admits a local algorithm with local complexity $O(\log n)$, extending the classification of Baire measurable colorings of Bernshteyn. A key ingredient of the proof is a new and simple characterization of local problems that can be solved in $O(\log n)$ rounds. We complement this result by showing separations between complexity classes from distributed computing, finitary factors, and descriptive combinatorics. Most notably, the class of problems that allow a distributed algorithm with sublogarithmic randomized local complexity is incomparable with the class of problems with a Borel solution. We hope that our treatment will help to view all three perspectives as part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn -- arXiv 2004.04905].