The computational complexity of Weisfeiler–Leman refinement

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

• Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence
• The complexity of identifying a graph by Weisfeiler–Leman refinement
• Unconditional time lower bounds for Weisfeiler–Leman equivalence
• Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs

Scope

The formalization gives the paper's parity-lift graph construction for every Weisfeiler–Leman dimension $k\ge4$, under both joint and separate conventions. From a finite choice system it constructs two equal-size uncolored graphs that are Weisfeiler–Leman equivalent exactly when the system has no successful compatible choice. The same equivalence holds for the pure variant.

A successful choice is detected by the specified early round: round two in the joint convention and round one in the separate convention. The graph-size bound is linear in the choice-system size with a factor depending on $k$. The paper's conditional running-time exclusions are outside these selected construction statements.

The formalization proves that deciding whether Weisfeiler–Leman refinement of a supplied dimension identifies a graph is EXPTIME-complete under polynomial-time many-one reductions. The input is a nonempty finite simple uncolored graph encoded by its adjacency matrix together with a positive binary-encoded dimension. Identification quantifies over every comparison graph, and the statement includes both the exponential-time upper bound and hardness.

The formalization proves an unconditional deterministic time lower bound for deciding $k$-dimensional Weisfeiler–Leman equivalence. For every sufficiently large fixed $k$, every deciding algorithm in the stated multitape Turing-machine or sequential logarithmic-word RAM model requires at least $n^{ck}$ worst-case time for all sufficiently large graph orders $n$, for one absolute $c>0$.

The bound covers both joint and separate replacement conventions and remains valid for explicit adjacency-matrix inputs restricted to simple connected uncolored graphs of diameter at most two.

The formalization proves EXPTIME-completeness of deciding joint-update $k$-dimensional Weisfeiler–Leman equivalence when $k\ge2$ is part of the input in binary and the graphs are encoded by explicit adjacency matrices. It proves both the general result and the restriction to connected simple uncolored graphs of equal positive order and maximum degree at most three. Completeness uses polynomial-time many-one reductions.

Comparator links