Existence of a Classical Polynomial-Time Exact Algorithm for the General Travelling Salesman Problem

Determine whether a deterministic classical algorithm exists that, given an arbitrary number of cities with an arbitrary symmetric or asymmetric cost matrix, computes an exact optimal Hamiltonian cycle in time polynomial in the number of cities (i.e., whether the general Travelling Salesman Problem admits a classical polynomial-time exact solution).

Background

The paper motivates quantum approaches to the Travelling Salesman Problem (TSP) by noting its NP-hardness and the limitations of classical methods. While exact classical algorithms exist, their worst-case runtime scales exponentially in the number of cities, and heuristic methods do not guarantee optimality. The authors explicitly state that classical polynomial-time algorithms are not known to solve the general TSP, highlighting a foundational unresolved question in computational complexity.

This open problem provides broader context for the work: the proposed single-qubit quantum algorithm aims to offer resource efficiency and potential speedups, but the existence of a classical polynomial-time exact algorithm for general TSP remains a central unresolved question.