Matroid-Based TSP Rounding for Half-Integral Solutions

TL;DR

This paper introduces a matroid-based rounding technique for half-integral solutions of the TSP relaxation, achieving improved approximation guarantees by leveraging max-entropy sampling and matroid intersection properties.

Contribution

It presents a novel rounding algorithm for TSP solutions that improves approximation factors using matroid intersection and max-entropy sampling methods.

Findings

Achieves less-than-1.5 approximation factor for TSP rounding

Utilizes matroid intersection polytope for improved guarantees

Builds on and enhances previous sampling-based approaches

Abstract

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, and a new use of max-entropy sampling, can give better guarantees.