In this paper we investigate instances with high integrality ratio of the subtour LP. We develop a procedure to generate families of Euclidean TSP instances whose integrality ratios converge to $\frac{4}{3}$ and may have a different structure than the instances currently known from the literature. Moreover, we compute the instances maximizing the integrality ratio for Rectilinear TSP with up to 10 vertices. Based on these instances we give families of instances whose integrality ratio converge to $\frac{4}{3}$ for Rectilinear, Multidimensional Rectilinear and Euclidean TSP that have similar structures. We show that our instances for Multidimensional Rectilinear TSP and the known instances for Metric TSP maximize the integrality ratio under certain assumptions. We also investigate the concept of local optimality with respect to integrality ratio and develop several algorithms to find instances with high integrality ratio. Furthermore, we describe a family of instances that are hard to solve in practice. The currently fastest TSP solver Concorde needs more than two days to solve an instance from the family with 52 vertices.
翻译：在本文中,我们调查了低水平LP高度整体性比率的事例。我们制定了一种程序,以产生欧洲-克利平原TSP案例的家庭,其整体性比率接近于$frac{4 ⁇ 3}美元,而且可能与文献中目前已知的情况结构不同。此外,我们还计算了使直线TSP整体性比率最大化的事例,最多达到10个顶脊椎。根据这些事例,我们向那些其整体性比率接近于$\frac{4 ⁇ 3}美元的情况的家庭提供了一些情况的家庭,这些家庭在Rectilinear、Mdolental Reclinear和Euclidean TSP方面有着类似的结构。我们表明,我们的多层次的TSP案例和Metritri TSP已知的情况在某些假设下最大限度地提高了整体性比率。我们还调查了当地整体性比率的最佳性概念,并制定了几种算法,以找到高度整体性比率的例子。此外,我们描述了一个难以在实践中解决的案件的家庭。目前最快的TSP Solentr Concordate需要两天以上的时间才能用52个家庭解决案例。
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