The multi-stripe travelling salesman problem

TL;DR

This paper introduces the q-stripe TSP, a generalization of the classical TSP that sums costs over multiple consecutive cities, analyzing its complexity and extending known theorems.

Contribution

It provides a complexity analysis of the q-stripe TSP for various structured matrices and generalizes Kalmanson's theorem to this new problem.

Findings

NP-hardness for certain matrix classes

Polynomial solvability in specific cases

Extension of Kalmanson's theorem

Abstract

In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q larger than 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP.