On approximate data reduction for the Rural Postman Problem: Theory and experiments

Given an undirected graph with edge weights and a subsetRof its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges ofR. We prove that RPP is WK[1]-complete parameterized by the number and weightdof edges traversed additionally to the required ones. Thus RPP instances cannot be polynomial-time compressed to instances of size polynomial indunless the polynomial-time hierarchy collapses. In contrast, denoting byb <= 2dthe number of vertices incident to an odd number of edges ofRand byc <= dthe number of connected components formed by the edges inR, we show how to reduce any RPP instanceIto an RPP instanceI(')with2b + O(c/epsilon)vertices inO(n(3))time so that any alpha-approximate solution forI(')gives an alpha(1 + epsilon)-approximate solution forI, for any alpha >= 1and epsilon > 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 parameterc.