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It Does Not Matter How You Define Locally Checkable Labelings

Authors Antonio Cruciani , Avinandan Das , Alesya Raevskaya , Jukka Suomela

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ACM Subject Classification
  • Theory of computation → Distributed computing models
Keywords
  • Locally checkable labeling problems
  • Distributed graph algorithms

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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. The definition strikes a careful balance between two objectives: the family of LCL problems is broad enough to capture 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.
Yet what we are seeing more and more often is that there are LCL problems with unexpected, counterintuitive properties: there are LCL problems that admit a distributed quantum advantage [SODA 2026] or benefit from shared randomness [ICALP 2025]; there is an LCL problem whose complexity in the LOCAL model depends on whether state transitions are restricted to computable functions [DISC 2026]; we can construct LCL problems with unnatural round complexities [STOC 2018]; and many questions related to the distributed complexity of LCL problems are undecidable [STOC 1993, PODC 2017]. Overall, the LCL problems seem to be a poor proxy for the family of natural local graph problems. With large volumes of research on LCL problems, the following questions are getting ever more pressing: Are we studying the right problem family? Maybe the above counterintuitive features are mere artifacts of the specific definition of LCLs introduced by Naor and Stockmeyer, and they disappear if we slightly restrict the family of LCL problems?
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 with little overhead; in particular, all counterintuitive properties listed above hold also for restricted LCLs.

Cite As Get BibTex

Antonio Cruciani, Avinandan Das, Alesya Raevskaya, and Jukka Suomela. It Does Not Matter How You Define Locally Checkable Labelings. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.10

Author Details

Antonio Cruciani
  • Aalto University, Espoo, Finland
Avinandan Das
  • Aalto University, Espoo, Finland
Alesya Raevskaya
  • Aalto University, Espoo, Finland
Jukka Suomela
  • Aalto University, Espoo, Finland

Funding

This work was supported in part by the Research Council of Finland, Grants 363558 and 359104.

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions, and Alkida Balliu, Dennis Olivetti, Sebastian Brandt and Henrik Lievonen for many fruitful discussions.

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