For a set Q of points in the plane and a real number δ ≥ 0 , let G δ ( Q ) be the graph defined on Q by connecting each pair of points at distance at most δ .We consider the connectivity of G δ ( Q ) 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 P of n - k points in the plane and a set S of k line segments in the plane, find the minimum δ ≥ 0 with the property that we can select one point p s ∈ s for each segment s ∈ S and the corresponding graph G δ ( P ∪ { p s ∣ s ∈ S } ) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in O ( f ( k ) n log n ) time, for a computable function f ( · ) . This implies that the problem is FPT when parameterized by k . The best previous algorithm uses O ( ( k ! ) k k k + 1 · n 2 k ) time and computes the solution up to fixed precision.