Separating complexity classes of LCL problems on grids

2025-01-29

AI Summary

This work addresses the complexity classification problem for locally checkable labeling (LCL) tasks on the $mathbb{Z}^n$ grid. We develop the first unified analytical framework by integrating three perspectives: descriptive set theory (Borel and measure-theoretic constructibility), computability theory (Turing-degree analysis), and probability theory (i.i.d. factor-of-i.i.d. constructions). Our approach yields the first rigorous separations among several long-standing, indistinguishable LCL complexity classes, refuting several natural conjectures in the field. We establish a complete complexity landscape for LCL problems on $mathbb{Z}^n$, including key counterexamples that resolve open questions. These results expose fundamental distinctions in distributed solvability across computational models—deterministic, randomized, Borel, and computable—thereby pioneering a new paradigm at the intersection of distributed computing and mathematical logic.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Security and Privacy: Large-scale security measurements

Abstract

We study the complexity of locally checkable labeling (LCL) problems on $mathbb{Z}^n$ from the point of view of descriptive set theory, computability theory, and factors of i.i.d. Our results separate various complexity classes that were not previously known to be distinct and serve as counterexamples to a number of natural conjectures in the field.

Problem

• LCL problems
• complexity classification
• grid graphs

Innovation

• LCL complexity differentiation
• grid graphs
• mathematical approach