[Paper] It does not matter how you define locally checkable labelings

Source: arXiv - 2602.18188v1

Overview

The paper It does not matter how you define locally checkable labelings shows that the classic definition of locally checkable labeling (LCL) problems—central to distributed graph algorithms—can be dramatically simplified without losing any expressive power. By proving that a very restricted “node‑edge‑checkable” formalism can be translated back and forth to the standard LCL model with only an (O(\log^* n)) overhead, the authors demonstrate that the whole distributed‑complexity landscape for LCLs is robust to the exact way we write the problem specification.

Key Contributions

• Equivalence proof between the traditional LCL definition and a minimal “node‑edge‑checkable” formalism on regular, unlabeled graphs.
• Bidirectional local reductions that require only a symmetry‑breaking oracle (e.g., a cheap coloring) and incur at most an additive (O(\log^* n)) rounds in the LOCAL model.
• Robustness insight: the expressive power of LCLs does not depend on the ability to reference global structures like short cycles.
• Clarification of the theory: simplifies the foundation for round‑elimination techniques and other meta‑algorithms used to classify LCL complexities.

Methodology

1. Problem Formalism – The authors start from the classic LCL definition (a finite set of allowed constant‑radius neighborhoods) and introduce a stripped‑down version where each node only checks constraints on its incident edges and the labels of its immediate neighbors (the “node‑edge‑checkable” model).
2. Symmetry‑Breaking Oracle – They assume access to a weak oracle that can break symmetry locally (e.g., a proper (O(\Delta^2)) coloring in (O(\log^* n)) rounds). This is a standard tool in distributed computing and adds only a negligible overhead.
3. Local Reductions – Two constructive reductions are given:
   • From any LCL to a node‑edge‑checkable problem, by encoding the original constant‑radius constraints into edge‑labels and local checks.
   • From a node‑edge‑checkable problem back to a standard LCL, by expanding the local view to simulate the edge‑centric checks.
4. Complexity Analysis – The reductions are shown to increase the round complexity by at most an additive (O(\log^* n)), which is essentially the cost of the symmetry‑breaking step.

Results & Findings

• Equivalence Established: Any LCL on regular, unlabeled graphs can be expressed as a node‑edge‑checkable problem and vice‑versa, with only a tiny additive overhead.
• No Need for Global Patterns: The ability to refer to structures like short cycles is unnecessary for capturing the full class of LCLs; local node‑edge constraints suffice.
• Implications for Classification: Existing classification results (e.g., the famous “(O(1)), (Θ(\log n)), (Θ(\log\log n)), (Θ(\log^* n))” hierarchy) remain valid under the simplified formalism, confirming that the theory’s foundations are stable.

Practical Implications

• Simpler Algorithm Design – Distributed system engineers can now write specifications using only node‑edge constraints, which are easier to reason about and implement in real code (e.g., in graph‑processing frameworks like Pregel or Giraph).
• Tooling & Verification – Automated verification tools can target the minimal formalism, reducing the state space they need to explore while still guaranteeing correctness for the full LCL class.
• Faster Prototyping – Since the only extra cost is an (O(\log^* n)) pre‑processing step (often negligible in practice), developers can prototype distributed graph algorithms without worrying about subtle definition details.
• Robust Meta‑Algorithms – Techniques such as round elimination, which are used to prove lower bounds, can be applied to a broader set of problems with confidence that the underlying model choice won’t affect the outcome.

Limitations & Future Work

• Regular Unlabeled Graphs – The equivalence is proved for regular graphs without node/edge labels; extending the result to irregular or already‑labeled graphs remains an open question.
• Symmetry‑Breaking Assumption – The reductions rely on a cheap symmetry‑breaking oracle; in environments where such coloring is expensive or impossible (e.g., highly dynamic networks), the overhead could be larger.
• Beyond LOCAL Model – The paper focuses on the LOCAL model of distributed computing. Investigating whether similar robustness holds in CONGEST or asynchronous settings is a promising direction.

Bottom line: Whether you describe a distributed graph problem with full‑blown LCL neighborhoods or with a lean node‑edge checklist, you get the same algorithmic power—making the theory both cleaner and more developer‑friendly.*

Authors

• Antonio Cruciani
• Avinandan Das
• Alesya Raevskaya
• Jukka Suomela

Paper Information

• arXiv ID: 2602.18188v1
• Categories: cs.DC
• Published: February 20, 2026
• PDF: Download PDF