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The Double Travelling Salesman Problem with Multiple Stacks, DTSPMS, dealswith the collect and delivery of n commodities in two distinct cities, wherethe pickup and the delivery tours are related by LIFO constraints. During thepickup tour, commodities are loaded into a container of k rows, or stacks, withcapacity c. This paper focuses on computational aspects of the DTSPMS, which isNP-hard.  We first review the complexity of two critical subproblems: deciding whethera given pair of pickup and delivery tours is feasible and, given a loadingplan, finding an optimal pair of pickup and delivery tours, are both polynomialunder some conditions on k and c.  We then prove a (3k)/2 standard approximation for the MinMetrickDTSPMS, wherek is a universal constant, and other approximation results for various versionsof the problem.  We finally present a matching-based heuristic for the 2DTSPMS, which is aspecial case with k=2 rows, when the distances are symmetric. This yields a1/2-o(1), 3/4-o(1) and 3/2+o(1) standard approximation for respectivelyMax2DTSPMS, its restriction Max2DTSPMS-(1,2) with distances 1 and 2, andMin2DTSPMS-(1,2), and a 1/2-o(1) differential approximation for Min2DTSPMS andMax2DTSPMS.

Approximation of the Double Travelling Salesman Problem with Multiple Stacks