A heuristic for the stacker crane problem on trees which is almost surely exact

Given an edge-weighted transportation network G and a list of transportation requests L, the stacker crane problem is to find a minimum-cost tour for a server along the edges of G that serves all requests. The server has capacity one, and starts and stops at the same vertex. In this paper, we consider the case that the transportation network G is a tree, and that the requests are chosen randomly according to a certain class of probability distributions. We show that a polynomial time algorithm by Frederickson and Guan [11], which guarantees a 4/3-approximation in the worst case, on almost all inputs finds a minimum-cost tour, along with a certificate of the optimality of its output.