Counting Locally Optimal Tours in the TSP

TL;DR

This paper proves that counting 2-optimal tours in TSP is #P-complete and provides bounds on their expected number in random instances, supported by numerical experiments and conjectures.

Contribution

It establishes the #P-completeness of counting 2-optimal TSP tours and offers bounds on their expected number in random instances.

Findings

Counting 2-optimal tours is #P-complete.

Expected number of 2-optimal tours is bounded by O(1.2098^n √n!).

Numerical experiments suggest the true bound is at most O(√n!).

Abstract

We show that the problem of counting the number of 2-optimal tours in instances of the Travelling Salesperson Problem (TSP) on complete graphs is #P-complete. In addition, we show that the expected number of 2-optimal tours in random instances of the TSP on complete graphs is $O(1.2098^n \sqrt{n!})$. Based on numerical experiments, we conjecture that the true bound is at most $O(\sqrt{n!})$, which is approximately the square root of the total number of tours.