On $$(1+\varepsilon )$$ -approximate Data Reduction for the Rural Postman Problem

Abstract

Given a graph \(G=(V,E)\) with edge weights and a subset \(R\subseteq E\) of required edges, the NP-hard Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. The number b of vertices incident to an odd number of edges of R and the number c of connected components formed by the edges in R are both bounded from above by the number of edges that has to be traversed additionally to the required ones. 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 \(\alpha \ge 1\) and \(\varepsilon >0\). That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We make first steps towards a PSAKS with respect to the parameter c.