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.

翻译：本文针对对称度量旅行商问题的经典Dantzig-Fulkerson-Johnson模型，研究其线性松弛——即子回路消除问题（SEP）的整数性间隙。该间隙被推测为4/3。我们证明，当求解包含n个节点的问题时，若最优SEP解至多包含n+6个非零分量，则该推测成立。为建立此结论，我们设计了一种结合理论分析与计算验证的新方法。