The Complexity of Maximizing the MST-ratio

Given a finite set of red and blue points in $\mathbb{R}^d$, the MST-ratio is the combined length of the Euclidean minimum spanning trees of red points and of blue points divided by the length of the Euclidean minimum spanning tree of the union of them. 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 the problem of finding the maximum MST-ratio of a given point set is NP-hard when the dimension is a part of the input. Moreover, we present a $O(n^2)$ running time $3$-approximation algorithm for it. As a 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 $n$ goes to infinity.