  In this article, we study the Euclidean minimum spanning tree problem in an
imprecise setup. The problem is known as the \emph{Minimum Spanning Tree
Problem with Neighborhoods} in the literature. We study the problem where the
neighborhoods are represented as non-crossing line segments. Given a set ${\cal
S}$ of $n$ disjoint line segments in $I\!\!R^2$, the objective is to find a
minimum spanning tree (MST) that contains exactly one end-point from each
segment in $\cal S$ and the cost of the MST is minimum among $2^n$ possible
MSTs. We show that finding such an MST is NP-hard in general, and propose a
$2\alpha$-factor approximation algorithm for the same, where $\alpha$ is the
approximation factor of the best-known approximation algorithm to compute a
minimum cost Steiner tree in an undirected graph with non-negative edge
weights. As an implication of our reduction, we can show that the unrestricted
version of the problem (i.e., one point must be chosen from each segment such
that the cost of MST is as minimum as possible) is also NP-hard. We also
propose a parameterized algorithm for the problem based on the "separability"
parameter defined for segments.
