It does not matter how you define locally checkable labelings

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).

💡 Research Summary

The paper investigates how robust the classic definition of Locally Checkable Labeling (LCL) problems is with respect to variations in the formalism. LCLs, introduced by Naor and Stockmeyer, are graph problems whose correctness can be verified by checking a finite set of local neighborhoods. This definition has been extremely fruitful: it captures many natural distributed tasks (coloring, MIS, maximal matching, sinkless orientation) while allowing strong complexity classifications (e.g., no LCL has complexity strictly between ω(log⁎ n) and o(log n) in the deterministic LOCAL model).

However, recent constructions have produced “artificial” LCLs exhibiting counter‑intuitive properties: quantum advantage, benefits from shared randomness or quantum states, dependence on computability of state transitions, exotic round complexities such as Θ(log 123.45 n) or Θ(n^0.12345), and even undecidable questions about their distributed complexity. These examples raise the suspicion that the Naor‑Stockmeyer definition may be too permissive, encompassing problems that are not “natural.”

To address this, the authors introduce a highly restricted formalism, called the RE (node‑edge) formalism, which is the standard “half‑edge” model used in round‑elimination. In the RE setting the input graph is a simple 3‑regular unlabeled graph; each half‑edge receives a label, and the only constraints are a node constraint (a multiset of three incident half‑edge labels) and an edge constraint (a multiset of the two half‑edge labels at the ends of an edge). Crucially, the problem description cannot directly refer to the existence of short cycles, node degrees, or any input labels; it is purely local and input‑free.

The main technical contribution is a two‑way local reduction between arbitrary LCLs and RE‑formalism problems, assuming the availability of a symmetry‑breaking oracle (e.g., a constant‑coloring that can be obtained in O(log⁎ n) rounds). The reduction proceeds through a sequence of intermediate problems:

1. A → B (regularization and input‑free encoding). Any LCL A can be transformed into an equivalent LCL B defined on input‑free 3‑regular graphs by using standard gadget constructions that encode original inputs into the structure of a regular graph.

2. B → C (PN‑checkable formulation). B is turned into a problem C whose correctness can be verified in the port‑numbering model, i.e., the verifier only sees tree‑like neighborhoods. This is achieved by adding a pair of auxiliary labels (x, y) to each node, interpreting them as a locally unique identifier; identical (x, y) pairs in a view are treated as the same logical node.

3. C → D (embedding into the RE formalism). The auxiliary identifiers are encoded as half‑edge labels, and the original node/edge constraints are expressed solely through the RE node‑ and edge‑constraints.

4. D → E (final RE problem). The resulting problem E is an RE‑formalism instance that is locally equivalent to the original LCL A.

Each step incurs at most a constant additive overhead, plus the O(log⁎ n) cost of breaking symmetry. Consequently, any LCL and its RE counterpart have the same distributed complexity up to an additive O(log⁎ n) term in the LOCAL model (and similarly in CONGEST, SLOCAL, online‑LOCAL, dynamic‑LOCAL, etc.).

From this equivalence the authors derive several corollaries: all previously known “counter‑intuitive” phenomena for LCLs also appear in the RE formalism. In particular, there exist RE‑formalism problems with quantum advantage, with shared‑randomness advantage, with undecidable complexity, and with exotic Θ(log n·c) or Θ(n^ε) round complexities. Thus these phenomena are intrinsic to the class of locally checkable problems, not artifacts of the original definition.

The paper concludes that the LCL definition is remarkably robust: even when severely restricted to input‑free, regular, half‑edge‑labelled graphs, the expressive power remains unchanged (modulo symmetry breaking). This validates the decades of research on LCLs as studying the “right” problem family. Open questions remain about the fine‑grained expressive gap when algorithms run in o(log⁎ n) rounds (the symmetry‑breaking region) and about the impact of further restricting the graph family (e.g., to trees) or allowing parallel edges and self‑loops. These directions point to a deeper understanding of locality, symmetry, and computational universality in distributed graph algorithms.