THE MAXIMUM TRAVELING SALESMAN PROBLEM ON BANDED MATRICES

Abstract

We investigate the Maximum Traveling Salesman Problem on banded distance matrices. A (p, q)-banded matrix has all its non-zero elements contained within a band consisting of p diagonals above the principal diagonal and q diagonals below it. We investigate the properties of the problem and prove that the number K of different permutations which can be optimal solutions for different instances of the problem, is exponential in n where n is the problem size (= the number of cities). For the Maximum-TSP on the (2, 0)-banded matrices, K=O(λn) where 1.7548 <λ< 1.7549, whereas on the (1, 1)-banded matrices K=O(λn) where 1.6180 <λ< 1.6181. Using recursive equations, we derive a linear-time algorithm that exactly solves the Maximum-TSP on the (2, 0)-banded distance matrices.