On the complexity of the multiple stack TSP, kSTSP

TL;DR

This paper investigates the computational complexity of the multiple Stack TSP (kSTSP), revealing how the arrangement of commodities and tour choices impact problem difficulty and providing insights into its NP-hard nature.

Contribution

It analyzes the complexity factors of kSTSP, showing polynomial-time solutions for fixed arrangements and highlighting the problem's NP-hardness and sensitivity to distance metrics.

Findings

Deciding tour compatibility is polynomial-time.

Optimal tours given an arrangement are polynomial-time computable but exponential in k.

Certain distance metrics can lead to arbitrarily suboptimal solutions.

Abstract

The multiple Stack Travelling Salesman Problem, STSP, deals with the collect and the deliverance of n commodities in two distinct cities. The two cities are represented by means of two edge-valued graphs (G1,d2) and (G2,d2). During the pick-up tour, the commodities are stored into a container whose rows are subject to LIFO constraints. As a generalisation of standard TSP, the problem obviously is NP-hard; nevertheless, one could wonder about what combinatorial structure of STSP does the most impact its complexity: the arrangement of the commodities into the container, or the tours themselves? The answer is not clear. First, given a pair (T1,T2) of pick-up and delivery tours, it is polynomial to decide whether these tours are or not compatible. Second, for a given arrangement of the commodities into the k rows of the container, the optimum pick-up and delivery tours w.r.t. this arrangement can be computed within a time that is polynomial in n, but exponential in k. Finally, we provide instances on which a tour that is optimum for one of three distances d1, d2 or d1+d2 lead to solutions of STSP that are arbitrarily far to the optimum STSP.