  We extend the notion of lossy kernelization, introduced by Lokshtanov et al.
[STOC 2017], to approximate Turing kernelization. An $\alpha$-approximate
Turing kernel for a parameterized optimization problem is a polynomial-time
algorithm that, when given access to an oracle that outputs $c$-approximate
solutions in $O(1)$ time, obtains an $(\alpha \cdot c)$-approximate solution to
the considered problem, using calls to the oracle of size at most $f(k)$ for
some function $f$ that only depends on the parameter.
  Using this definition, we show that Independent Set parameterized by
treewidth $\ell$ has a $(1+\varepsilon)$-approximate Turing kernel with
$O(\frac{\ell^2}{\varepsilon})$ vertices, answering an open question posed by
Lokshtanov et al. [STOC 2017]. Furthermore, we give
$(1+\varepsilon)$-approximate Turing kernels 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
$(1+\varepsilon)$-approximate Turing kernels of polynomial size when
parameterized by treewidth. We use this to obtain approximate Turing kernels
for Vertex-Disjoint $H$-packing for connected graphs $H$, Clique Cover,
Feedback Vertex Set and Edge Dominating Set.
