[Paper Review] Distributed Algorithms, the Lovász Local Lemma, and Descriptive Combinatorics

Anton Bernshteyn|arXiv (Cornell University)|Apr 10, 2020

Advanced Topology and Set Theory4 citations

TL;DR

This paper establishes a bridge between distributed algorithms and descriptive combinatorics by showing that efficient distributed coloring algorithms on finite graphs yield well-behaved Borel, measurable, or Baire-measurable colorings on infinite Borel graphs of bounded degree. It further proves a measurable and Baire-measurable version of the symmetric Lovász Local Lemma under a stronger condition, enabling new results on measurable chromatic numbers in graphs without short cycles.

ABSTRACT

In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or measure. To this end, we show that such well-behaved colorings can be produced using certain powerful techniques from finite combinatorics and computer science. First, we prove that efficient distributed coloring algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of bounded degree; roughly speaking, deterministic algorithms produce Borel colorings, while randomized algorithms give measurable and Baire-measurable colorings. Second, we establish measurable and Baire-measurable versions of the Symmetric Lovász Local Lemma (under the assumption $\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}$, which is stronger than the standard LLL assumption $\mathsf{p}(\mathsf{d} + 1) \leq e^{-1}$ but still sufficient for many applications). From these general results, we derive a number of consequences in descriptive combinatorics and ergodic theory.

Motivation & Objective

• To establish sufficient conditions under which combinatorial colorings on Borel graphs can be made well-behaved in topological or measure-theoretic senses.
• To transfer techniques from finite combinatorics—specifically distributed algorithms and the Lovász Local Lemma—into the descriptive combinatorics setting.
• To derive new quantitative bounds on measurable chromatic numbers for Borel graphs with bounded degree and restricted girth.
• To prove a measurable and Baire-measurable version of the symmetric Lovász Local Lemma under a stronger-than-usual condition.
• To unify algorithmic and descriptive set-theoretic approaches to coloring problems in infinite combinatorics.

Proposed method

• Adapts the LOCAL model of distributed computing to show that deterministic distributed algorithms produce Borel colorings, while randomized ones yield measurable and Baire-measurable colorings on Borel graphs.
• Introduces a Borel-reduction framework to simulate randomized distributed algorithms in the measurable and Baire-measurable settings via probabilistic constructions on finite sets of partial solutions.
• Applies a refined analysis of the Lovász Local Lemma by imposing a stronger condition: $\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}$, which ensures the existence of measurable and Baire-measurable solutions.
• Uses recursive construction of partial solutions with nonmeager or conull domains to ensure global solutions inherit regularity (measurability or Baire-measurability).
• Employs Borel reductions and probabilistic averaging arguments to lift finite combinatorial results (e.g., from CSPs) to infinite structures with regularity.
• Leverages results from descriptive set theory—such as comeager sets and Borel determinacy—to ensure the final coloring functions are measurable or Baire-measurable.

Experimental results

Research questions

• RQ1 Can efficient distributed algorithms on finite graphs be used to construct well-behaved colorings (Borel, measurable, or Baire-measurable) on infinite Borel graphs of bounded degree?
• RQ2 Under what conditions does a measurable or Baire-measurable version of the symmetric Lovász Local Lemma hold in the context of standard Borel spaces?
• RQ3 What are the tight bounds on the measurable chromatic number of Borel graphs with bounded degree and no short cycles?
• RQ4 How close can the measurable chromatic number be to the combinatorial chromatic number in Borel graphs under degree and girth constraints?
• RQ5 Can the gap between combinatorial and measurable chromatic numbers be closed for graphs with large girth, and what role does the Lovász Local Lemma play in this?

Key findings

• For every $\Delta \geq \Delta_0$, if $c \leq \sqrt{\Delta + 1/4} - 5/2$, then the measurable chromatic number of a Borel graph of maximum degree $\leq \Delta$ equals its combinatorial chromatic number whenever the latter is at most $\Delta - c$.
• The bound $c \leq \sqrt{\Delta + 1/4} - 5/2$ is within 2 of optimal, as shown by a proposition in the paper.
• For graphs with no 3-cycles, the $\mu$-measurable chromatic number is at most $(4 + \varepsilon)\Delta / \log \Delta$ for sufficiently large $\Delta$.
• For graphs with no 4-cycles, the $\mu$-measurable chromatic number is at most $(1 + \varepsilon)\Delta / \log \Delta$ for sufficiently large $\Delta$, matching the known lower bound up to a constant factor.
• The measurable chromatic number of acyclic Borel graphs is at least $(1/2 + o(1))\Delta / \log \Delta$, showing that the bounds in the paper are optimal up to constants.
• The symmetric Lovász Local Lemma can be adapted to the measurable and Baire-measurable settings under the condition $\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}$, which is stronger than the classical $\mathsf{p}(\mathsf{d}+1) \leq e^{-1}$ but still sufficient for many applications.