Parameterized computation of strong and weak coloring numbers

Determine whether there exists an algorithm that computes the strong r-coloring number and weak r-coloring number of an n-vertex graph in time n^{f(r,col_r(G))} and n^{f(r,wcol_r(G))}, respectively, for some function f, for every positive integer r at least 2.

Background

The paper notes that exact computation of weak coloring numbers is NP-complete for r at least 3, while r-admissibility can be computed efficiently on bounded-expansion classes. This problem asks whether the strong and weak coloring numbers nevertheless admit algorithms whose exponent depends only on r and the value of the parameter being computed.

References

Let $r\geq 2$ be a positive integer. Does there exist an algorithm that, given an $n$-vertex graph $G$, computes $\col_r(G)$ and $\wcol_r(G)$ in time $n{f(r,\col_r(G))}$, or $n{f(r,\wcol_r(G))}$, for some function~$f$?

— On the generalized coloring numbers  (2501.08698 - Siebertz, 15 Jan 2025) in Problem, Section 3.2, 'Bounding r-admissibility'