Approximation Algorithms for the Min–Max Mixed Rural Postmen Cover Problem and Its Variants

Abstract

In this work, we introduce a multi-vehicle (or multi-postman) extension of the classical Mixed Rural Postman Problem, which we call the Min–Max Mixed Rural Postmen Cover Problem (MRPCP). The MRPCP is defined on a mixed graph \(G=(V,E,A)\), where V is the vertex set, E denotes the (undirected) edge set and A represents the (directed) arc set. Let \(F\subseteq E\) (\(H\subseteq A\)) be the set of required edges (required arcs). There is a nonnegative weight associated with each edge and arc. The objective is to determine no more than k closed walks to cover all the required edges in F and all the required arcs in H such that the weight of the maximum weight closed walk is minimized. By replacing closed walks with (open) walks in the MRPCP, we obtain the Min–Max Mixed Rural Postmen Walk Cover Problem (MRPWCP). The Min–Max Mixed Chinese Postmen Cover Problem (MCPCP) is a special case of the MRPCP where \(F=E\) and \(H=A\). The Min–Max Stacker Crane Cover Problem (SCCP) is another special case of the MRPCP where \(F=\emptyset \) and \(H=A\) For the MRPCP with the input graph satisfying the weakly symmetric condition, i.e., for each arc there exists a parallel edge whose weight is not greater than this arc, we devise a \(\frac{27}{4}\)-approximation algorithm. This algorithm achieves an approximation ratio of \(\frac{33}{5}\) for the SCCP with the weakly symmetric condition. Moreover, we obtain the first 5-approximation algorithm (4-approximation algorithm) for the MRPWCP (MCPCP) with the weakly symmetric condition.