Hamiltonian Graphs and the Traveling Salesman Problem

TL;DR

This paper introduces a new polynomial-time algorithm for solving the Traveling Salesman Problem using a novel characterization of Hamiltonian graphs and an ordered weighted adjacency list, with extensions to quantum algorithms for search problems.

Contribution

It presents a new characterization of Hamiltonian graphs and an exact polynomial-time algorithm for TSP, along with quantum algorithms for unstructured search and K-SAT problems.

Findings

The algorithm finds shortest Hamiltonian paths and circuits efficiently.

A new characterization of shortest Hamiltonian tours under triangle inequality.

Quantum algorithms significantly improve search capabilities in NP-complete problems.

Abstract

A new characterization of Hamiltonian graphs using f-cutset matrix is proposed. Based on this new characterization, a new exact polynomial time algorithm for the traveling salesman problem (TSP) is developed. We then define the so-called ordered weighted adjacency list for given weighted complete graph and proceed to the paper's main result, namely, the exact algorithm based on the utilization of the ordered weighted adjacency list and the simple properties that any path or circuit must satisfy. This algorithm performs checking of sub-lists, containing (p-1) entries (edge pairs) for paths and p entries (edge pairs) for circuits, chosen from ordered adjacency list in a well defined sequence to determine exactly the shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph of p vertices. The procedure has intrinsic advantage of landing on the desired solution in quickest possible time and even in worst case in polynomial time. A new characterization of the shortest Hamiltonian tour for a weighted complete graph satisfying triangle inequality (i.e. for tours passing through every city on a realistic map of cities where cities can be taken as points on a Euclidean plane) is also proposed. Finally, we propose a classical algorithm for unstructured search, three new quantum algorithms for unstructured search, which exponentially speed up the searching ability in the unstructured database, and one quantum algorithm for solving a K-SAT problem and indicate its effect on traveling salesman problem and other NP-complete problems.