Nonexistence of a Universal Algorithm for Traveling Salesman Problems in Constructive Mathematics

TL;DR

This paper proves that there is no universal algorithm capable of solving all instances of the traveling salesman problem within the framework of constructive mathematics, highlighting fundamental limitations in algorithmic solutions.

Contribution

It provides a formal proof of the nonexistence of a universal algorithm for TSP in constructive mathematics, covering both symmetric and asymmetric cases.

Findings

No universal algorithm exists for symmetric TSP.

No universal algorithm exists for asymmetric TSP.

The proof applies within the framework of constructive mathematics.

Abstract

Proposed initially from a practical circumstance, the traveling salesman problem caught the attention of numerous economists, computer scientists, and mathematicians. These theorists were instead intrigued by seeking a systemic way to find the optimal route. Many attempts have been made along the way and all concluded the nonexistence of a general algorithm that determines optimal solution to all traveling salesman problems alike. In this study, we present proof for the nonexistence of such an algorithm for both asymmetric (with oriented roads) and symmetric (with unoriented roads) traveling salesman problems in the setup of constructive mathematics.