The subtour relaxation of the traveling salesman problem (TSP) plays a central role in approximation algorithms and polyhedral studies of the TSP. A long-standing conjecture asserts that the integrality gap of the subtour relaxation for the metric TSP is exactly 4/3. In this paper, we extend the exact verification of this conjecture for small numbers of vertices. Using the framework introduced by Benoit and Boyd in 2008, we confirm their results up to n=10. We further show that for n=11 and n=12, the published lists of extreme points of the subtour polytope are incomplete: one extreme point is missing for n=11 and twenty-two extreme points are missing for n=12. We extend the enumeration of the extreme points of the subtour polytope to instances with up to 14 vertices in the general case. Restricted to half-integral vertices, we extend the enumeration of extreme points up to n=17. Our results provide additional support for the 4/3-Conjecture. Our lists of extreme points are available on the public bonndata repository (https://doi.org/10.60507/FK2/JK95PC).

翻译：旅行商问题（TSP）的子回路松弛在TSP的近似算法和多面体研究中扮演核心角色。一个长期存在的猜想断言：度量TSP子回路松弛的整数间隙恰好为4/3。本文将该猜想在顶点数较小时的确切验证范围进行了扩展。借助Benoit与Boyd于2008年提出的框架，我们将其结果确认至n=10。进一步，我们发现对于n=11和n=12，已公布的子回路多面体极值点列表并不完整：n=11时缺少一个极值点，n=12时缺少二十二个极值点。我们将一般情形下子回路多面体极值点的枚举范围扩展至含14个顶点的问题实例；针对半整数顶点，我们将极值点的枚举范围扩展至n=17。我们的结果为4/3猜想提供了进一步支撑。极值点列表已公开于bonndata存储库（https://doi.org/10.60507/FK2/JK95PC）。