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

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 ≤ 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).