Approximate Turing kernelization for problems parameterized by treewidth

We extend the notion of lossy kernelization, introduced by Lokshtanov et al. (2017) [19], to approximate Turing kernelization. An α-approximate Turing kernelization for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs c-approximate solutions in $O\left(1\right)$ time, computes an $\alpha \cdot c$-approximate solution to the considered problem, using calls to the oracle of size at most $f\left(k\right)$ for some function f that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth ℓ has a $(1+\epsilon )$-approximate Turing kernelization with $O\left(\frac{{\ell }^2}{\epsilon }\right)$ vertices, answering an open question posed by Lokshtanov et al. (2017) [19]. Furthermore, we give $(1+\epsilon )$-approximate Turing kernelizations for the following graph problems parameterized by treewidth: Vertex Cover, Edge Clique Cover, Edge-Disjoint Triangle Packing, and Connected Vertex Cover. We generalize the result for Independent Set and Vertex Cover by showing that all graph problems that we will call friendly admit $(1+\epsilon )$-approximate Turing kernelizations of polynomial size when parameterized by treewidth. We use this to establish approximate Turing kernelizations for Vertex-Disjoint H-packing for connected graphs H, Clique Cover, Feedback Vertex Set, and Edge Dominating Set.