Connectivity with Uncertainty Regions Given as Line Segments

For a set \({\mathcal {Q}}\) of points in the plane and a real number \(\delta \ge 0\), let \({\mathbb {G}}_\delta ({\mathcal {Q}})\) be the graph defined on \({\mathcal {Q}}\) by connecting each pair of points at distance at most \(\delta \).We consider the connectivity of \({\mathbb {G}}_\delta ({\mathcal {Q}})\) 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 \({\mathcal {P}}\) of \(n-k\) points in the plane and a set \({\mathcal {S}}\) 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 \(s\in {\mathcal {S}}\) and the corresponding graph \({\mathbb {G}}_\delta ( {\mathcal {P}}\cup \{ p_s\mid s\in {\mathcal {S}}\})\) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in \({{\,\mathrm{{\mathcal {O}}}\,}}(f(k) n \log n)\) time, for a computable function \(f(\cdot )\). This implies that the problem is FPT when parameterized by k. The best previous algorithm uses \({{\,\mathrm{{\mathcal {O}}}\,}}((k!)^k k^{k+1}\cdot n^{2k})\) time and computes the solution up to fixed precision.