A linear time algorithm for the -neighbour Travelling Salesman Problem on a Halin graph and extensions

The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamiltonian cycle in an edge-weighted graph, where costs are defined for all pairs of edges contained in the Hamiltonian cycle. The problem is shown to be strongly NP-hard on a Halin graph. We also consider a variation of the QTSP, called the k-neighbour TSP (TSP(k)). Two edges e and f , e 6= f , are k-neighbours on a tour τ if and only if a shortest path (with respect to the number of edges) between e and f along τ and containing both e and f , has exactly k edges, for k ≥ 2. In (TSP(k)), a fixed nonzero cost is considered for a pair of distinct edges in the cost of a tour τ only when the edges are p-neighbours on τ for 2 ≤ p ≤ k. We give a linear time algorithm to solve TSP(k) on a Halin graph for k = 3, extending existing algorithms for the cases k = 1, 2. Our algorithm can be extended further to solve TSP(k) in polynomial time on a Halin graph with n nodes when k = O(logn). The possibility of extending our results to some fully reducible class of graphs is also discussed. TSP(k) can be used to model the Permuted Variable Length Markov Model in bioinformatics as well as an optimal routing problem for unmanned aerial vehicles (UAVs).