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Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence
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Abstract
For k ≥ 4, we construct two uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when a prescribed finite-domain choice system has no compatible choice. The system has $k+1$ domains for joint refinement and k for separate-coordinate refinement. A successful choice is detected after two joint rounds or one separate round. Applied to sparse satisfiability, the reduction gives fixed-dimension $n^{\Omega(k)}$ time exclusions under positive-rate ETH, even for deciding equality of these early histograms.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Radu Curticapean, Mingjun Liu, Daniel Neuen, and Tim Seppelt, Lower Bounds for Weisfeiler–Leman (Zenodo:10.5281/zenodo.23213174, 2026-10-07). This concurrent AI-assisted manuscript cites the parity-lift paper and gives lower bounds on the size of bounded-treewidth distinguishing homomorphism witnesses; its preface dates the AI-assisted results to September 19–29 and describes the manuscript as a hastily written public timestamp. [AI-generated per source] [family-level]
Formalization
- OpenAI, The computational complexity of Weisfeiler–Leman refinement: Lean scope notes (OpenAI Math Release, lean/docs/133.md, 2026-10-06). The release's Lean scope note covers the parity-lift characterization for every $k\ge4$ under the joint and separate conventions (comparator
ParityLifts.lean), early-round detection and graph-size bounds, and states that the conditional (ETH) running-time consequences are outside the formalized statements. [AI-generated per source] [family-level] - Radu Curticapean, Mingjun Liu, Daniel Neuen, and Tim Seppelt, Lean formalizations of Weisfeiler–Leman results (supplement to Zenodo:10.5281/zenodo.23213174, 2026-10-07). The archive documents WL/game/homomorphism correspondence, round bounds and a deterministic register compiler, with decision-complexity reductions outside its scope. [AI-generated per source] [family-level]
Related prior literature mentioned by others
- Martin Grohe, Moritz Lichter, Daniel Neuen, and Pascal Schweitzer, Compressing CFI Graphs and Lower Bounds for the Weisfeiler–Leman Refinements (arXiv:2308.11970v2, 2025-01-27; Journal of the ACM 72(3), 2025). Curticapean and coauthors cite its Ω(n^{k/2}) refinement-round bound as a predecessor to their concurrent work on this family. [family-level]
News and blogs
- Tim Seppelt, News: Lower Bounds for Weisfeiler–Leman (author website, 2026-10-07). Seppelt announces the concurrent manuscript in response to OpenAI’s release and reports extensions involving classical complexity, parameterized complexity and iteration bounds. [family-level]
Elsewhere
- Emergent Mind: family 133
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026,
author = {{OpenAI}},
title = {{Parity lifts and bounded-treewidth witnesses for Weisfeiler--Leman equivalence}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf}{OAI:Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/133.md. The Lean statements have not been checked by people against the paper's claims.
Family 133: The computational complexity of Weisfeiler–Leman refinement
Proves unconditional $n^{\Omega(k)}$ deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.
Other papers in this family:
- The complexity of identifying a graph by Weisfeiler–Leman refinement · September 25, 2026
- Unconditional time lower bounds for Weisfeiler–Leman equivalence · September 25, 2026
- Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs · September 25, 2026