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Container terminals are exchange hubs that interconnect many transportation modes and facilitate the flow of containers. Among other elements, terminals include a yard which serves as temporary storage space. In the yard, containers are piled up by cranes to form blocks of stacks. During the shipment process, containers are carried from the stacks to ships following a given sequence. Hence, if a high priority container is placed below low priority ones, such obstructing containers have to be moved (relocated) to other stacks. Given a set of stacks and a retrieval sequence, the aim in the Pre-marshalling Problem (pmp) is to sort the initial configuration according to the retrieval sequence using a minimum number of relocations, so that no new relocations are needed during the shipment. The objective in the Block Relocation Problem (brp) is to retrieve all the containers according to the retrieval sequence by using a minimum number of relocations. This paper presents a new unified integer programming model for solving the pmp, the brp, and the Restricted brp (r-brp) variant. The new formulations are compared with existing mathematical models for these problems, as well as with other exact methods that combines combinatorial lower bounds and the branch-and-bound (B&B) framework, by using a large set of instances available in the literature. The numerical experiments * Corresponding author: Tel. +33 1 4940 4082.show that the proposed models are able to outperform the approaches based on mathematical programming. Nevertheless, the B&B algorithms achieve the best results both in terms of computation time and number of instances solved to optimality.

We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst-and the best-value solutions of an instance. We next show that the 2 OPT, one of the mostknown traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2 OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, for any > 0, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 + and traveling salesman with distances 1 and 2 within better than 741/742 + . Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now.

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any > 0, it is NP-hard to approximate both problems better than within 741/742 + . The same results hold when dealing with a generalization of min and max TSP12, where instead of 1 and 2, edges are valued by a and b.

Abstract. Given a universal constant k, the multiple Stack Travelling Salesman Problem (kSTSP in short) consists in finding a pickup tour T 1 and a delivery tour T 2 of n items on two distinct graphs. The pickup tour successively stores the items at the top of k containers, whereas the delivery tour successively picks the items at the current top of the containers: thus, the couple of tours are subject to LIFO ("Last In First Out") constraints. This paper aims at finely characterizing the complexity of kSTSP in regards to the complexity of TSP. First, we exhibit tractable sub-problems: on the one hand, given two tours T 1 and T 2 , deciding whether T 1 and T 2 are compatible can be done within polynomial time; on the other hand, given an ordering of the n items into the k containers, the optimal tours can also be computed within polynomial time. Note that, to the best of our knowledge, the only family of combinatorial precedence constraints for which constrained TSP has been proven to be in P is the one of PQ-trees, [2]. Finally, in a more prospective way and having in mind the design of approximation algorithms, we study the relationship between optimal value of different TSP problems and the optimal value of kSTSP.

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We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any > 0, it is NP-hard to approximate both problems better than within 741/742 + . The same results hold when dealing with a generalization of min and max TSP12, where instead of 1 and 2, edges are valued by a and b.

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