# Connectivity with Uncertainty Regions Given as Line Segments

## TL;DR

This paper presents a new algorithm to efficiently solve a complex geometric connectivity problem involving uncertain point locations.

## Contribution

The paper introduces an exact fixed-parameter tractable algorithm for computing connectivity with uncertain points represented as line segments.

## Key findings

- An algorithm is developed with time complexity O(f(k) n log n) for computing optimal connectivity.
- The problem is proven to be fixed-parameter tractable when parameterized by k.
- The new algorithm outperforms previous methods with significantly reduced computational complexity.

## Abstract

For a set \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}$$\end{document}Q of points in the plane and a real number \documentclass[12pt]{minimal}
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\begin{document}$$\delta \ge 0$$\end{document}δ≥0, let \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {G}}_\delta ({\mathcal {Q}})$$\end{document}Gδ(Q) be the graph defined on \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}$$\end{document}Q by connecting each pair of points at distance at most \documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}δ.We consider the connectivity of \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {G}}_\delta ({\mathcal {Q}})$$\end{document}Gδ(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 \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}$$\end{document}P of \documentclass[12pt]{minimal}
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\begin{document}$$n-k$$\end{document}n-k points in the plane and a set \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {S}}$$\end{document}S of k line segments in the plane, find the minimum \documentclass[12pt]{minimal}
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\begin{document}$$\delta \ge 0$$\end{document}δ≥0 with the property that we can select one point \documentclass[12pt]{minimal}
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\begin{document}$$p_s\in s$$\end{document}ps∈s for each segment \documentclass[12pt]{minimal}
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\begin{document}$$s\in {\mathcal {S}}$$\end{document}s∈S and the corresponding graph \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {G}}_\delta ( {\mathcal {P}}\cup \{ p_s\mid s\in {\mathcal {S}}\})$$\end{document}Gδ(P∪{ps∣s∈S}) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in \documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{{\mathcal {O}}}\,}}(f(k) n \log n)$$\end{document}O(f(k)nlogn) time, for a computable function \documentclass[12pt]{minimal}
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\begin{document}$$f(\cdot )$$\end{document}f(·). This implies that the problem is FPT when parameterized by k. The best previous algorithm uses \documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{{\mathcal {O}}}\,}}((k!)^k k^{k+1}\cdot n^{2k})$$\end{document}O((k!)kkk+1·n2k) time and computes the solution up to fixed precision.