It does not matter how you define locally checkable labelings

2026-02-20

AI Summary

This work investigates the equivalence between restricted formulations of locally checkable labeling (LCL) problems—such as node-edge-checkable instances—and their general counterparts. By introducing a technique of local reductions augmented with a symmetry-breaking oracle, the authors establish bidirectional transformations between the two forms in the LOCAL model. They demonstrate that even under restrictions that preclude direct reference to structural features like short cycles, LCL problems retain their full expressive power. The central contribution lies in proving that these two formulations are equivalent in terms of distributed complexity: mutual reductions incur only an additive O(log* n) rounds of communication overhead. This result reveals a remarkable robustness of LCL problems with respect to their definitional form, underscoring that their computational complexity is invariant under natural syntactic restrictions.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

Locally checkable labeling problems (LCLs) form the foundation of the modern theory of distributed graph algorithms. First introduced in the seminal paper by Naor and Stockmeyer [STOC 1993], these are graph problems that can be described by listing a finite set of valid local neighborhoods. This seemingly simple definition strikes a careful balance between two objectives: they are a family of problems that is broad enough so that it captures numerous problems that are of interest to researchers working in this field, yet restrictive enough so that it is possible to prove strong theorems that hold for all LCL problems. In particular, the distributed complexity landscape of LCL problems is now very well understood. In this work we show that the family of LCL problems is extremely robust to variations. We present a very restricted family of locally checkable problems (essentially, the"node-edge checkable"formalism familiar from round elimination, restricted to regular unlabeled graphs); most importantly, such problems cannot directly refer to e.g. the existence of short cycles. We show that one can translate between the two formalisms (there are local reductions in both directions that only need access to a symmetry-breaking oracle, and hence the overhead is at most an additive $O(\log^* n)$ rounds in the LOCAL model).

Problem

• Locally Checkable Labelings
• Distributed Graph Algorithms
• Complexity Landscape
• Local Reductions
• Symmetry-Breaking

Innovation

• Locally Checkable Labelings
• distributed graph algorithms
• round elimination
• LOCAL model
• symmetry-breaking oracle