The Double Travelling 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 MinMetrickDTSPMS, 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 2DTSPMS, 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 Max2DTSPMS, its restriction Max2DTSPMS-(1,2) with distances 1 and 2, and Min2DTSPMS-(1,2), and a 1/2-o(1) differential approximation for Min2DTSPMS and Max2DTSPMS.

中文翻译：

---

多栈双旅行商问题的逼近

多栈双旅行商问题，DTSPMS，处理在两个不同城市的 n 个商品的收集和交付，其中取货和交付旅行通过 LIFO 约束相关联。在接送过程中，商品被装入一个容量为 c 的 k 行或堆垛的容器中。本文重点介绍 DTSPMS 的计算方面，这是 NP-hard。我们首先回顾两个关键子问题的复杂性：确定给定的一对取货和送货路线是否可行，以及给定装载计划，找到一对最佳的取货和送货路线，在 k 和 c 的某些条件下都是多项式的。然后，我们证明了 MinMetrickDTSPMS 的 (3k)/2 标准近似值，其中 k 是一个通用常数，以及该问题各种版本的其他近似结果。我们最终为 2DTSPMS 提出了一种基于匹配的启发式方法，这是当距离对称时 k=2 行的特殊情况。这分别为 Max2DTSPMS 产生了 1/2-o(1)、3/4-o(1) 和 3/2+o(1) 标准近似值，其限制为 Max2DTSPMS-(1,2)，距离为 1 和 2，和 Min2DTSPMS-(1,2)，以及 Min2DTSPMS 和 Max2DTSPMS 的 1/2-o(1) 差分近似。