Approximate Turing kernelization and lower bounds for domination problems

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.