Gilmore–Gomory type traveling salesman problems

Abstract

One of the well-known, solvable cases of the traveling salesman problem (TSP) is the Gilmore–Gomory case. A generalization of their algorithm to what we call the GG scheme is known to solve a fairly large subclass of the problem. In this paper, we identify new classes of TSP for which the GG scheme produces an optimal solution and which properly include the class of TSP for which the cost matrix is a permuted distribution matrix. Implementation of the GG scheme is NP-hard for these classes of TSP. However, we identify some subclasses for which the GG scheme can be implemented in polynomial time. Some of these new solvable cases are proper generalizations of some well-known cases. Scope and purpose Since TSP is NP-hard, it is interesting to identify, generalize and unify polynomially testable and polynomially solvable subclasses of the TSP. Perhaps, the most well-known solvable case of TSP is the one identified by Gilmore and Gomory in 1964. We show that their approach can be generalized and applied to a fairly larger class of TSP. We point out some polynomially testable classes of TSP for which the approach can be implemented in polynomial time. Some of these new solvable cases generalize and unify some well-known cases.