Approximation of the Double Traveling Salesman Problem with Multiple Stacks

• Published in: Theoretical computer science Vol. 877; pp. 74 - 89
• Main Authors: Alfandari, Laurent, Toulouse, Sophie
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 20.07.2021
• Subjects: Approximation; Complexity; Matching algorithms; Traveling salesman problem; Traveling salesman problem; Approximation; Matching algorithms; Complexity
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2021.05.016

•We study the approximation of the Double Traveling Salesman Problem with Multiple Stacks.•We study the complexity of two important subproblems.•We provide approximation results for both standard and differential approximation.•Approximation results are derived from reductions to the TSP in the general case.•Algorithms based on optimal matchings and completions are analyzed for the case with two stacks. The Double Traveling Salesman Problem with Multiple Stacks, DTSPMS, deals with the collect and delivery of n commodities in two distinct cities, where the pickup and the delivery tours are related by LIFO constraints. During the pickup tour, commodities are loaded into a container of k rows, or stacks, with capacity c. This paper focuses on computational aspects of the DTSPMS, which is NP-hard. We first review the complexity of two critical subproblems: deciding whether a given pair of pickup and delivery tours is feasible and, given a loading plan, finding an optimal pair of pickup and delivery tours, are both polynomial under some conditions on k and c. We then prove a (3k)/2 standard approximation for the Min Metric k DTSPMS, where k is a universal constant, and other approximation results for various versions of the problem. We finally present a matching-based heuristic for the 2 DTSPMS, which is a special case with k=2 rows, when the distances are symmetric. This yields a 1/2−o(1), 3/4−o(1) and 3/2+o(1) standard approximation for respectively Max 2 DTSPMS, its restriction Max 2 DTSPMS(1,2) with distances 1 and 2, and Min 2 DTSPMS(1,2), and a 1/2−o(1) differential approximation for Min 2 DTSPMS and Max 2 DTSPMS.