Solving a Class of Traveling Salesman Problems Analytically

Abstract

This paper addresses a class of Traveling Salesman Problems (TSP) in which a route must be made to a series of nodes and return to the original location and attempts to solve it using analytical methods. The problem will be presented as a matrix of routes, much as might be seen in a national road map, excepting for there being in this case less entries. This familiar arrangement of routes will be cast as a matrix problem and solved using familiar formulations of quadratic forms. This solution, should it prove successful, can be contrasted with differing numeric or even iterative methods, such as the well-known Gomory cut method of solving integer linear programs. The advantage, should it prove tenable, will be theoretic in that a familiar and accessible form of quadratic forms can be readily applied to the problem and to similar cases.