  For $\alpha > 1$, an $\alpha$-approximate (bi-)kernel is a polynomial-time
algorithm that takes as input an instance $(I, k)$ of a problem $\mathcal{Q}$
and outputs an instance $(I',k')$ (of a problem $\mathcal{Q}'$) of size bounded
by a function of $k$ such that, for every $c\geq 1$, a $c$-approximate solution
for the new instance can be turned into a $(c\cdot\alpha)$-approximate solution
of the original instance in polynomial time. This framework of lossy
kernelization was recently introduced by Lokshtanov et al. We study Connected
Dominating Set (and its distance-$r$ variant) parameterized by solution size on
sparse graph classes like biclique-free graphs, classes of bounded expansion,
and nowhere dense classes. We prove that for every $\alpha>1$, Connected
Dominating Set admits a polynomial-size $\alpha$-approximate (bi-)kernel on all
the aforementioned classes. Our results are in sharp contrast to the
kernelization complexity of Connected Dominating Set, which is known to not
admit a polynomial kernel even on $2$-degenerate graphs and graphs of bounded
expansion, unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complement our
results by the following conditional lower bound. We show that if a class
$\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for
some value of $r\in\mathbb{N}$ there cannot exist an $\alpha$-approximate
bi-kernel for the (Connected) Distance-$r$ Dominating Set problem on
$\mathcal{C}$ for any $\alpha>1$ (assuming the Gap Exponential Time
Hypothesis).
