Connectivity with uncertainty regions given as line segmentsCabello, Sergio (Avtor) Gajser, David (Avtor) computational geometryuncertaintygeometric optimizationfixed parameter tractabilityparametric searchFor 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 ${\mathcal O}(f(k) n \log n)$ time, for a computable function $f(\cdot)$. This implies that the problem is FPT when parameterized by $k$. The best previous algorithm uses ${\mathcal O}((k!)^k k^{k+1}\cdot n^{2k})$ time and computes the solution up to fixed precision.20242024-05-13 10:15:17Neznano18913sl