  An $\alpha$-approximate polynomial Turing kernelization is a polynomial-time
algorithm that computes an $(\alpha c)$-approximate solution for a
parameterized optimization problem when given access to an oracle that can
compute $c$-approximate solutions to instances with size bounded by a
polynomial in the parameter. Hols et al. [ESA 2020] showed that a wide array of
graph problems admit a $(1+\varepsilon)$-approximate polynomial Turing
kernelization when parameterized by the treewidth of the graph and left open
whether Dominating Set also admits such a kernelization.
  We show that Dominating Set and several related problems parameterized by
treewidth do not admit constant-factor approximate polynomial Turing
kernelizations, even with respect to the much larger parameter vertex cover
number, under certain reasonable complexity assumptions.On the positive side,
we show that all of them do have a $(1+\varepsilon)$-approximate polynomial
Turing kernelization for every $\varepsilon>0$ for the joint parameterization
by treewidth and maximum degree, a parameter which generalizes cutwidth, for
example.
