# Layers and Matroids for the Traveling Salesman's Paths

## TL;DR

This paper presents a layered convex combination approach of matroid bases to analyze solutions of the TSP path problem, providing a new combinatorial algorithm and polyhedral insights.

## Contribution

It introduces a layered matroid-based proof and a strongly polynomial algorithm for decomposing TSP path solutions, enhancing understanding and solution methods.

## Key findings

- Convex combination of generalized Gao-trees for TSP paths
- A new combinatorial algorithm for layered decomposition
- Polyhedral insights into TSP path solutions

## Abstract

Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive "generalized Gao-trees". We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the $s-t$ path TSP.