Connectivity with uncertainty regions given as line segments

Abstract: For a set $Q$ of points in the plane and a real number $\delta \ge 0$, let $\mathbb{G}\delta(Q)$ be the graph defined on $Q$ by connecting each pair of points at distance at most $\delta$. We consider the connectivity of $\mathbb{G}\delta(Q)$ 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 $\delta\ge 0$ with the property that we can select one point $p_s\in s$ for each segment $s\in S$ and the corresponding graph $\mathbb{G}_\delta ( P\cup { p_s\mid s\in S})$ 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 $f(\cdot)$. This implies that the problem is FPT when parameterized by $k$. The best previous algorithm uses $O((k!)^k k^{k+1}\cdot n^{2k})$ time and computes the solution up to fixed precision.