A Linear Time Algorithm for the $3$-neighbour Traveling Salesman Problem on Halin graphs and extensions

TL;DR

This paper introduces a linear time algorithm for solving the 3-neighbour TSP on Halin graphs, extending to broader graph classes and applications like UAV routing and scheduling.

Contribution

It presents the first linear time algorithm for TSP(3) on Halin graphs and extends the approach to fully reducible graph classes for fixed k.

Findings

Linear time algorithm for TSP(3) on Halin graphs

Extension to polynomial time for fully reducible graphs

Applications in UAV routing and machine scheduling

Abstract

The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamilton cycle in an edge-weighted graph, where costs are defined on all pairs of edges contained in the Hamilton cycle. This is a more general version than the commonly studied QTSP which only considers pairs of adjacent edges. We define a restricted version of QTSP, the $k$-neighbour TSP (TSP($k$)), and give a linear time algorithm to solve TSP($k$) on a Halin graph for $k\leq 3$. This algorithm can be extended to solve TSP($k$) on any fully reducible class of graphs for any fixed $k$ in polynomial time. This result generalizes corresponding results for the standard TSP. TSP($k$) can be used to model various machine scheduling problems as well as an optimal routing problem for unmanned aerial vehicles (UAVs).