Is a LOCAL algorithm computable?

2026-02-24

AI Summary

This work investigates whether node state updates in the LOCAL model of distributed computing should be restricted to computable functions and how this restriction affects the round complexity of Locally Checkable Labeling (LCL) problems. By integrating techniques from distributed computing theory, computability analysis, and LCL complexity classification, the paper establishes for the first time that the round complexity of LCL problems is jointly determined by the computability assumption and prior knowledge of the graph size $n$. The central contribution is the construction of an LCL problem $\Pi$ that requires $\Omega(\sqrt{n})$ rounds in the computable LOCAL model without knowledge of $n$, yet can be solved in $O(\log n)$ rounds either when an upper bound on $n$ is known or when non-computable updates are permitted. Moreover, the authors demonstrate that this complexity separation is not isolated but exhibits a general phenomenon.

Technology Category

• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Common definitions of the"standard"LOCAL model tend to be sloppy and even self-contradictory on one point: do the nodes update their state using an arbitrary function or a computable function? So far, this distinction has been safe to neglect, since problems where it matters seem contrived and quite different from e.g. typical local graph problems studied in this context. We show that this question matters even for locally checkable labeling problems (LCLs), perhaps the most widely studied family of problems in the context of the LOCAL model. Furthermore, we show that assumptions about computability are directly connected to another aspect already recognized as highly relevant: whether we have any knowledge of $n$, the size of the graph. Concretely, we show that there is an LCL problem $\Pi$ with the following properties: 1. $\Pi$ can be solved in $O(\log n)$ rounds if the \textsf{LOCAL} model is uncomputable. 2. $\Pi$ can be solved in $O(\log n)$ rounds in the computable model if we know any upper bound on $n$. 3. $\Pi$ requires $\Omega(\sqrt{n})$ rounds in the computable model if we do not know anything about $n$. We also show that the connection between computability and knowledge of $n$ holds in general: for any LCL problem $\Pi$, if you have any bound on $n$, then $\Pi$ has the same round complexity in the computable and uncomputable models.

Problem

• LOCAL model
• computability
• locally checkable labeling
• distributed complexity
• graph size knowledge

Innovation

• LOCAL model
• computability
• locally checkable labeling
• distributed complexity
• graph size knowledge