Travelling salesman paths on Demidenko matrices

TL;DR

This paper identifies a new polynomially solvable case of the Path-TSP when the distance matrix is a Demidenko matrix, using combinatorial properties and a dynamic programming approach.

Contribution

It introduces a novel polynomial-time solution for Path-TSP on Demidenko matrices, expanding the class of efficiently solvable instances.

Findings

Polynomial-time algorithm for Path-TSP on Demidenko matrices

Key combinatorial properties of optimal solutions

Dynamic programming with O(n^6) complexity

Abstract

In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t. In this paper we identify a new polynomially solvable case of the Path-TSP where the distance matrix of the cities is a so-called Demidenko matrix. We identify a number of crucial combinatorial properties of the optimal solution, and we design a dynamic program with time complexity $O(n^6)$.