Approximation Algorithms for Some Min-Max and Minimum Stacker Crane Cover Problems

Abstract

We study two stacker crane cover problems and their variants. Given a mixed graph \(G=(V,E,A)\) with vertex set V, edge set E and arc set A. Each edge or arc is associated with a nonnegative weight. The Min-Max Stacker Crane Cover Problem (SCC) aims to find at most k closed walks covering all the arcs in A such that the maximum weight of the closed walks is minimum. The Minimum Stacker Crane Cover Problem (MSCC) is to cover all the arcs in A by a minimum number of closed walks of length at most \(\lambda \). The Min-Max Stacker Crane Walk Cover Problem (SCWC)/Minimum Stacker Crane Walk Cover Problem (MSCWC) is a variant of the SCC/MSCC problem with closed walks replaced by (open) walks.For the SCC problem with symmetric arc weights, i.e. for every arc there is a parallel edge of no greater weight, we obtain a 33/5-approximation algorithm. This improves on the previous 37/5-approximation algorithm for a restricted case of the SCC problem with symmetric arc weights. If the arc weights are symmetric, we devise the first constant-factor approximation algorithms for the SCWC problem, the MSCC problem and the MSCWC problem with ratios 5, 5 and 7/2, respectively. Finally, for the (general) MSCWC problem we first propose a 4-approximation algorithm.