Approximating bottleneck spanning trees on partitioned tuples of points

We present approximation algorithms for the following NP-hard optimization problems related to bottleneck spanning trees in metric spaces. 1. The disjoint bottleneck spanning tree problem: Given $n$ pairs of points in a metric space, find two disjoint trees each containing exactly one point from each pair and minimize the largest edge length (over all edges of both trees). It is known that approximating this problem by a factor better than 2 is NP-hard. We present a 4-approximation algorithm for this problem. This improves upon the previous best known approximation ratio of $9$. Our algorithm extends to a $(3k-2)$-approximation for a more general case where points are partitioned into $k$-tuples and we seek $k$ disjoint trees. 2. The generalized bottleneck spanning tree problem: Given $n$ points in some metric space that are partitioned into clusters of size at most 2, find a tree that contains exactly one point from each cluster and minimizes the largest edge length. We show that it is NP-hard to approximate this problem by a factor better than 2, and present a 3-approximation algorithm. 3. The partitioned bottleneck spanning tree problem: Given $kn$ points in some metric space, find $k$ trees each containing exactly $n$ points and minimize the largest edge length (over all edges of the $k$ trees). We show that it is NP-hard to approximate this problem by a factor better than 2 for any $k\ge 2$. We present an $\alpha$-approximation algorithm for this problem where $\alpha=2$ for $k=2,3$ and $\alpha=3$ for $k\ge 4$. Towards obtaining these approximation ratios we present tight upper bounds on the edge lengths of $k$ equal-size disjoint trees that can be obtained from the nodes of a given tree. This result is of independent interest. Our hardness proofs imply that it is NP-hard to approximate the non-metric version of the above problems within any constant factor.