  Given an undirected graph with edge weights and a subset $R$ of its edges,
the Rural Postman Problem (RPP) is to find a closed walk of minimum total
weight containing all edges of $R$. We prove that RPP is WK[1]-complete
parameterized by the number and cost $d$ of edges traversed additionally to the
required ones. Thus, in particular, RPP instances cannot be polynomial-time
compressed to instances of size polynomial in $d$ unless the polynomial-time
hierarchy collapses. In contrast, denoting by $b\leq 2d$ the number of vertices
incident to an odd number of edges of $R$ and by $c\leq d$ the number of
connected components formed by the edges in $R$, we show how to reduce any RPP
instance $I$ to an RPP instance $I'$ with $2b+O(c/\varepsilon)$ vertices in
$O(n^3)$ time so that any $\alpha$-approximate solution for $I'$ gives an
$\alpha(1+\varepsilon)$-approximate solution for $I$, for any $\alpha\geq 1$
and $\varepsilon>0$. That is, we provide a polynomial-size approximate
kernelization scheme (PSAKS). We experimentally evaluate it on wide-spread
benchmark data sets as well as on two real snow plowing instances from Berlin.
On instances with few connected components, the number of vertices and required
edges is reduced to about $50\,\%$ at a $1\,\%$ solution quality loss. We also
make first steps towards a PSAKS for the parameter $c$.
