Summary: 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 \(\mathrm{WK}[1]\)-complete parameterized by the number and weight \(d\) of edges traversed additionally to the required ones. Thus 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 \(I^\prime\) with \(2b + O(c/ \varepsilon)\) vertices in \(O(n^3)\) time so that any \(\alpha\)-approximate solution for \(I^\prime\) 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. We also make first steps toward a PSAKS for the parameter \(c\).
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