Matroid-based TSP rounding for half-integral solutions

We show how to round any half-integral solution to the subtour-elimination relaxation for the TSP, while losing a less-than - 1.5 factor. Such a rounding algorithm was recently given by Karlin, Klein, and Oveis Gharan based on sampling from max-entropy distributions. We build on an approach of Haddadan and Newman to show how sampling from the matroid intersection polytope, combined with a novel use of max-entropy sampling, can give better guarantees.