On the Complexity of the Multiple Stack TSP, kSTSP

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.