Connectivity with Uncertainty Regions Given as Line Segments

For a set $$ of points in the plane and a real number $$, let $$ be the graph defined on $$ by connecting each pair of points at distance at most $$.We consider the connectivity of $$ in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set $$ of $$ points in the plane and a set $$ of k line segments in the plane, find the minimum $$ with the property that we can select one point $$ for each segment $$ and the corresponding graph $$ is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in $$ time, for a computable function $$. This implies that the problem is FPT when parameterized by k. The best previous algorithm uses $$ time and computes the solution up to fixed precision.