Computing the integrality gap of the asymmetric travelling salesman problem

This extended abstract outlines the results of our investigations into the integrality gap of the Asymmetric Travelling Salesman Problem for small values of n. Specifically, we have computed the exact integrality gap for 4 ≤ n ≤ 7 and we have found a lower bound on the integrality gap for 8 ≤ n ≤ 15. We have also created a new family of extreme points based on our data for which the integrality gaps approach 2 as n → ∞ and furthermore, this family achieves a larger gap for specific values of n than a previously known family.