The one-commodity pickup and delivery travelling salesman problem on a path or a tree

Optimization algorithms for both path and tree topology classes of the one-commodity pickup and delivery travelling salesman problem (1-PDTSP) are proposed in this article, which focus on minimizing the route distance to transport products among pickup and delivery customers by a single vehicle with a limited capacity of k. Each pickup customer provides one unit volume of the product while each delivery customer requires one unit volume of the product. For the path case, we propose an O(n2/min (k, n)) algorithm for any arbitrary k, and two O(n) algorithms for k = 1 and k = ∞. For the tree case, O(n2) and O(n) algorithms are proposed for k = 1 and k = ∞, respectively. Moreover, when k is arbitrary, the problem becomes NP-hard in the strong sense.