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.

翻译：对于平面上的点集 $Q$ 和实数 $\delta \ge 0$，设 $\mathbb{G}_\delta(Q)$ 为在 $Q$ 上定义的图，其中每对距离不超过 $\delta$ 的点之间均存在连接。我们考虑在少数点的位置存在不确定性时 $\mathbb{G}_\delta(Q)$ 在最优场景下的连通性，但已知每个不确定点位于某条线段内。更精确地，我们考虑以下优化问题：给定平面上的 $n-k$ 个点集 $P$ 和平面上的 $k$ 条线段集 $S$，求最小的 $\delta\ge 0$，使得对于每条线段 $s\in S$，我们可选择一点 $p_s\in s$，且相应的图 $\mathbb{G}_\delta ( P\cup \{ p_s\mid s\in S\})$ 是连通的。已知该问题是 NP-困难的。我们提供一种算法，可在 $O(f(k) n \log n)$ 时间内精确计算最优解，其中 $f(\cdot)$ 为可计算函数。这表明该问题在参数 $k$ 下是 FPT 的。此前最优算法的时间复杂度为 $O((k!)^k k^{k+1}\cdot n^{2k})$，且仅能计算到固定精度的解。