Complexity of variable-dimensional Weisfeiler–Leman equivalence
Follow the latest progress by email.
What is the complexity of deciding joint-update k-dimensional Weisfeiler–Leman equivalence when k≥2 is binary encoded as part of the input? Graphs are explicit adjacency matrices, rather than succinct encodings.
Why it matters and what counts as progress
Why it matters: Fixed-dimension polynomial refinement does not determine complexity when dimension can grow with the input.
What counts as progress: A matching upper and hardness classification under this encoding, including any restrictions such as connected equal-order uncolored subcubic graphs.
How the problem developed
- 1968 · Foundations [1]
Boris Weisfeiler and Andrei Leman Introduced the graph-refinement method that later became the Weisfeiler–Leman family.
- 1992 · Lower bound [1]
Jin-Yi Cai, Martin Fürer and Neil Immerman Proved linear lower bounds on the dimension needed to distinguish certain nonisomorphic graphs, showing fixed dimension does not settle general isomorphism.
- 25 Sept 2026 · Reported claim [1]
OpenAI The preprint proves EXPTIME-completeness for deciding joint-update k-WL equivalence when k is binary-encoded, including connected graphs of maximum degree three.
AI contribution
Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs
For joint-update k-dimensional Weisfeiler–Leman equivalence, with k≥2 binary encoded as part of the input and graphs given by explicit adjacency matrices, the test is EXPTIME-complete. The hardness also holds for connected equal-order simple uncolored graphs of maximum degree three.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI proves variable-dimension Weisfeiler–Leman equivalence EXPTIME-complete