The multi-stripe travelling salesman problem

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 ≥ 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

Another look at the shoelace TSP : the case of very old shoes

What is the most efficient way of lacing a shoe? Mathematically speaking, this question concerns the structure of certain special cases of the bipartite travelling salesman problem (BTSP). We show that techniques developed for the analysis of the (standard) TSP may be applied successfully to characterize well-solvable cases of the BTSP and the shoelace problem. In particular, we present a polynomial time algorithm that decides whether there exists a renumbering of the cities such that the resulting distance matrix carries a benevolent combinatorial structure that allows one to write down the optimal solution without further analysis of input data. Our results generalize previously published well-solvable cases of the shoelace problem. Keywords: Bipartite travelling salesman problem; shoelace problem; polynomially solvable case; relaxed Monge matrix; pick-and-place robo

One-sided monge TSP is NP-Hard

The Travelling Salesman Problem (TSP) is a classical NP-hard optimisation problem. There exist, however, special cases of the TSP that can be solved in polynomial time. Many of the well-known TSP special cases have been characterized by imposing special four-point conditions on the underlying distance matrix. Probably the most famous of these special cases is the TSP on a Mange matrix, which is known to be polynomially solvable (as are some other generally NP-hard problems restricted to this class of matrices). By relaxing the four-point conditions corresponding to Monge matrices in different ways, one can define other interesting special cases of the TSP, some of which turn out to be polynomially solvable, and some NP-hard. However, the complexity status of one such relaxation, which we call one-sided Monge TSP (also known as the TSP on a relaxed Supnick matrix), has remained unresolved. In this note, vie show that this version of the TSP problem is NP-hard. This completes the full classification of all possible four-point conditions for symmetric TSP