扩展度量TSP的子游走松弛整数规划间隙计算
Extending Exact Integrality Gap Computations for the Metric TSP
AI总结：
本文通过扩展子游走松弛的整数规划间隙验证，发现n=11和n=12时极端点列表不完整，并将极端点枚举扩展到14个顶点，为4/3猜想提供支持。
摘要：
旅行商问题（TSP）的子游走松弛在近似算法和多面体研究中起着核心作用。一个长期存在的猜想认为，度量TSP的子游走松弛的整数规划间隙恰好为4/3。本文扩展了该猜想对小顶点数的精确验证。使用Benoit和Boyd于2008年引入的框架，我们验证了n=10的结果。进一步显示，对于n=11和n=12，已发表的子游走多面体极端点列表不完整：n=11缺失一个极端点，n=12缺失22个极端点。我们扩展了子游走多面体极端点的枚举，一般情况下扩展到最多14个顶点。在半整数顶点受限的情况下，将极端点枚举扩展到n=17。我们的结果为4/3猜想提供了额外支持。我们的极端点列表可在公共bonndata仓库（https://doi.org/10.60507/FK2/JK95PC）获取。
英文摘要：
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 to a larger number 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 $n=16$ vertices in the general case. Restricted to half-integral extreme points, we extend the enumeration to $n=18$. 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).
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