Connectivity with Uncertainty Regions Given as Line Segments

Abstract

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