On the Complexity of the Multiple Stack TSP, kSTSP

Abstract

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.

New formulation and branch‐and‐cut algorithm for the pickup and delivery traveling salesman problem with multiple stacks

“…() to ensure a capacity and LIFO feasible solution. Indeed, in the particular case in which all items are picked up before any delivery is made, the authors in Toulouse and Wolfler Calvo () and Casazza et al. () show that the problem is NP‐hard.…”

“…No polynomial time algorithm is expected for solving the packing problem solved in Côté et al (2012a) to ensure a capacity and LIFO feasible solution. Indeed, in the particular case in which all items are picked up before any delivery is made, the authors in Toulouse and Wolfler Calvo (2009) and Casazza et al (2012) show that the problem is NP-hard. In order to facilitate the identification of LIFO and capacity constraints violated by a tour, besides the natural design variables x ∈ B |A| , we consider a new set of variables to better model the LIFO and capacity constraints of each stack.…”

Approximability of the Multiple Stack TSP

“…; if on the opposite STSP assumes n stacks, then it reduces to two independent TSP (packings that store one commodity per stack do not induce any LIFO constraint). The litterature on computational aspects of STSP is rather thin: see [1,2] for metaheuristic approachs, [3] for an exact approach (that solves 2STSP via the computation of the k best tours in I A , I B ), [4] for an analysis of STSP complexity. We here make use of natural connections to TSP and observations made in [4] in order to locate STSP within the approximation hierarchy.…”

“…One the one hand, one could wonder on what part of the problem does structure / impact the most the solution / its complexity. It appeared in [4] that both the packing and the tour subproblems are tractable:…”

A branch‐and‐bound algorithm for the double travelling salesman problem with two stacks

“…An interesting analysis of basic subproblems of the DTSPMS has been carried out in Refs. [5] and [24]. To the best of our knowledge, only two exact solution approaches for the DTSPMS have been presented [20, 21].…”