Computational complexity of the Weisfeiler-Leman dimension
CoRR(2024)
Abstract
The Weisfeiler-Leman dimension of a graph G is the least number k such
that the k-dimensional Weisfeiler-Leman algorithm distinguishes G from
every other non-isomorphic graph. The dimension is a standard measure of the
descriptive complexity of a graph and recently finds various applications in
particular in the context of machine learning. In this paper, we study the
computational complexity of computing the Weisfeiler-Leman dimension. We
observe that in general the problem of deciding whether the Weisfeiler-Leman
dimension of G is at most k is NP-hard. This is also true for the more
restricted problem with graphs of color multiplicity at most 4. Therefore, we
study parameterized and approximate versions of the problem. We give, for each
fixed k≥ 2, a polynomial-time algorithm that decides whether the
Weisfeiler-Leman dimension of a given graph of color multiplicity at most 5
is at most k. Moreover, we show that for these color multiplicities this is
optimal in the sense that this problem is P-hard under logspace-uniform
AC_0-reductions. Furthermore, for each larger bound c on the color
multiplicity and each fixed k ≥ 2, we provide a polynomial-time
approximation algorithm for the abelian case: given a relational structure with
abelian color classes of size at most c, the algorithm outputs either that
its Weisfeiler-Leman dimension is at most (k+1)c or that it is larger than
k.
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