Given a finite set of red and blue points in $\Rspace^d$, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The maximum MST-ratio of a point set is the maximum MST-ratio over all non-trivial colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time $3$-approximation algorithm for this problem. As part of the proof, we show that, in any metric space, the maximum MST-ratio is smaller than $3$. Additionally, we study the average MST-ratio over all colorings of a set of $n$ points. We show that this average is always at least $\frac{n-2}{n-1}$, and for $n$ random points uniformly distributed in a $d$-dimensional unit cube, the average tends to $\sqrt[d]{2}$ in expectation as $n$ approaches infinity.

翻译：给定 $\Rspace^d$ 中一个有限的红点与蓝点集合，MST比率定义为红点集合的欧几里得最小生成树总长度与蓝点集合的欧几里得最小生成树总长度之和，除以红蓝点并集的欧几里得最小生成树长度。点集的最大MST比率是指在其所有非平凡红蓝着色方案中MST比率的最大值。我们证明了当维度作为输入的一部分时，寻找给定点集的最大MST比率是NP难的。此外，我们提出了一个针对该问题的二次时间3-近似算法。作为证明的一部分，我们证明了在任何度量空间中，最大MST比率均小于3。另外，我们研究了n个点所有着色方案的平均MST比率。我们证明该平均值始终不小于 $\frac{n-2}{n-1}$，且对于均匀分布在d维单位立方体中的n个随机点，当n趋于无穷时，其期望平均MST比率趋近于 $\sqrt[d]{2}$。