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

In this paper, we introduce the Min-Max Mixed Rural Postmen Cover Problem (MRPCP), which is an extension of the Mixed Rural Postman Problem to the situation where several postmen (or vehicles) are available. Given a mixed graph G = (V, E, A) with vertex set V, (undirected) edge set E, (directed) arc set A. Each edge and arc is associated with a nonnegative weight (or length). The MRPCP is to find at most k closed walks to cover a set F subset of E of required edges and a set H subset of A of required arcs. The goal is to minimize the maximum weight of the closed walks. We also study two variants of the MRPCP. The first one is the Min-Max Mixed Rural Postmen Walk Cover Problem (MRPWCP) in which the closed walks are replaced by (open) walks. The second one is called the Min-Max Mixed Chinese Postmen Cover Problem (MCPCP), which is a special case of the MRPCP where F = E and H = A. If the input graph satisfies the weakly symmetric condition, i.e. for every arc there is a parallel edge of no greater weight, we propose an algorithm for the MRPCP, whose approximation ratio lies between 33/5 and 27/4 depending on the ratio of the weight of H to that of F. When F = empty set and H = A, it is a 33/5 -approximation algorithm for the MinMax Stacker Crane Cover Problem (SCCP). In addition, we devise the first constant-factor approximation algorithms for the MRPWCP and the MCPCP with ratios 5 and 4, respectively, when the input graphs satisfy the weakly symmetric condition.