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.

翻译：一个$\alpha$近似多项式图灵核化是一种多项式时间算法，当给定一个能对规模受参数多项式有界的实例计算$c$近似解的预言机时，它能为参数化优化问题计算$(\alpha c)$近似解。Hols等人[ESA 2020]表明，当以图的树宽为参数时，大量图问题允许$(1+\varepsilon)$近似多项式图灵核化，并公开了支配集是否也允许此类核化的问题。我们证明，在合理的复杂性假设下，以树宽为参数时，支配集及相关问题不允许常数因子近似多项式图灵核化，即使针对更大的参数顶点覆盖数也是如此。积极方面，我们表明，对于由树宽和最大度——这一参数可推广如割宽——构成的联合参数化，所有这些问题对于每个$\varepsilon>0$都确实具有$(1+\varepsilon)$近似多项式图灵核化。