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 compute exactly 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 is using $O((k!)^k k^{k+1}\cdot n^{2k})$ time and computes the solution up to fixed precision.

翻译：对于平面点集 $Q$ 和实数 $\delta \ge 0$，定义图 $\mathbb{G}_\delta(Q)$ 为连接所有距离不超过 $\delta$ 的点对所得的图。我们考虑在少数点位置不确定但已知每个不确定点所在线段的情形下，$\mathbb{G}_\delta(Q)$ 连通性的最优配置。具体而言，研究如下优化问题：给定平面上的 $n-k$ 个点组成的集合 $P$ 和 $k$ 条线段组成的集合 $S$，求最小的 $\delta\ge 0$，使得存在选择为每条线段 $s\in S$ 选取一点 $p_s\in s$ 后，对应图 $\mathbb{G}_\delta ( P\cup \{ p_s\mid s\in S\})$ 连通。已知该问题为NP难问题。我们提出算法在 $O(f(k) n \log n)$ 时间内精确求解最优解，其中 $f(\cdot)$ 为可计算函数，这表明该问题关于参数 $k$ 具有固定参数可解性。此前最优算法需 $O((k!)^k k^{k+1}\cdot n^{2k})$ 时间且仅能求解固定精度下的解。