# Simulated annealing approach to verify vertex adjacencies in the traveling salesperson polytope

## TL;DR

This paper presents a heuristic simulated annealing method to verify nonadjacency of vertices in the traveling salesperson polytope, addressing an NP-complete problem with practical testing on random tours.

## Contribution

It introduces a novel simulated annealing heuristic for verifying vertex nonadjacency in TSP polytopes, leveraging cycle covers and matchings.

## Key findings

- Algorithm always correctly identifies nonadjacent vertices.
- Tested successfully on random and pyramidal Hamiltonian tours.
- Provides a practical approach to a known NP-complete problem.

## Abstract

We consider 1-skeletons of the symmetric and asymmetric traveling salesperson polytopes whose vertices are all possible Hamiltonian tours in the complete directed or undirected graph, and the edges are geometric edges or one-dimensional faces of the polytope. It is known that the question whether two vertices of the symmetric or asymmetric traveling salesperson polytopes are nonadjacent is NP-complete. A sufficient condition for nonadjacency can be formulated as a combinatorial problem: if from the edges of two Hamiltonian tours we can construct two complementary Hamiltonian tours, then the corresponding vertices of the traveling salesperson polytope are not adjacent. We consider a heuristic simulated annealing approach to solve this problem. It is based on finding a vertex-disjoint cycle cover and a perfect matching. The algorithm has a one-sided error: the answer "not adjacent" is always correct, and was tested on random and pyramidal Hamiltonian tours.