The Integrality Gap of the Traveling Salesman Problem is $4/3$ if the LP Solution Has at Most $n+6$ Non-zero Components

We address the classical Dantzig - Fulkerson - Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution has at most n + 6 non-zero components, then the conjecture is true. To establish this result, we devise a new methodology that combines theoretical analysis and computational verification.