It does not matter how you define locally checkable labelings

TL;DR

This paper demonstrates that the class of locally checkable labeling problems (LCLs) is highly robust to formal variations, showing equivalences between different problem formalisms with minimal overhead in distributed algorithms.

Contribution

The authors prove that a restricted family of LCL problems can be translated to and from the general formalism with only a small overhead, highlighting the robustness of the LCL framework.

Findings

Most restricted LCL problems cannot directly specify short cycles.

There exist local reductions between different LCL formalisms with $O( ext{log}^* n)$ overhead.

The LCL framework is highly adaptable to various problem specifications.

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).