Another Look at the Shoelace TSP: The Case of Very Old Shoes

摘要

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.