One way to speed up the calculation of optimal TSP tours in practice is eliminating edges that are certainly not in the optimal tour as a preprocessing step. In order to do so several edge elimination approaches have been proposed in the past. In this work we investigate two of them in the scenario where the input consists of $n$ independently distributed random points in the 2-dimensional unit square with bounded density function from above and below by arbitrary positive constants. We show that after the edge elimination procedure of Hougardy and Schroeder the expected number of remaining edges is $\Theta(n)$, while after that the non-recursive part of Jonker and Volgenant the expected number of remaining edges is $\Theta(n^2)$.

翻译：在实际计算最优TSP环游的过程中，一种加速方法是作为预处理步骤，消除那些肯定不会出现在最优环游中的边。为此，过去已提出了多种边消除方法。本文研究了其中两种方法在输入由二维单位正方形内$n$个独立同分布随机点组成、且密度函数上下界均为任意正常数的情形下的表现。我们证明，经过Hougardy和Schroeder的边消除步骤后，剩余边数的期望为$\Theta(n)$；而经过Jonker和Volgenant的非递归部分处理后，剩余边数的期望为$\Theta(n^2)$。