| Title: | Connectivity with uncertainty regions given as line segments |
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| Authors: | ID Cabello, Sergio (Author)
ID Gajser, David (Author) |
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| Files: |  RAZ_Cabello_Sergio_2024.pdf (685,93 KB)
MD5: DB4D9F8EFC927B763D18C6AFE0557AC9
 http://dx.doi.org/10.1007/s00453-023-01200-5
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| Language: | English |
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| Work type: | Scientific work |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FNM - Faculty of Natural Sciences and Mathematics
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| Abstract: | For a set $\mathcal{Q}$ of points in the plane and a real number $δ$ ≥ 0, let $\mathbb{G}_δ(\mathcal{Q})$ be the graph defined on $\mathcal{Q}$ by connecting each pair of points at distance at most $δ$. We consider the connectivity of $\mathbb{G}_δ(\mathcal{Q})$ 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 $δ$ ≥ 0 with the property that we can select one point $p_s$ ∈ $s$ for each segment $s$ ∈ $\mathcal{S}$ and the corresponding graph $\mathbb{G}_δ(\mathcal{P} ∪ \{p_s$ | $s ∈ \mathcal{S}\})$ is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in $\mathcal{O}( f (k)n$ ${\rm log}$ $n)$ time, for a computable function $f$ (·). This implies that the problem is FPT when parameterized by $k$. The best previous algorithm uses $\mathcal{O}((k!)^kk^{k+1} · n^{2k})$ time and computes the solution up to fixed precision. |
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| Keywords: | computational geometry, uncertainty, geometric optimization, fixed parameter tractability, parametric search |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Submitted for review: | 17.03.2023 |
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| Article acceptance date: | 11.12.2023 |
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| Publication date: | 09.01.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 1512-1544 |
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| Numbering: | Letn. 6, št. 5 |
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| PID: | 20.500.12556/DKUM-88567  |
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| UDC: | 519.17 |
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| ISSN on article: | 0178-4617 |
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| COBISS.SI-ID: | 180364547 |
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| DOI: | 10.1007/s00453-023-01200-5 |
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| Publication date in DKUM: | 21.10.2025 |
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| Views: | 110 |
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| Downloads: | 6 |
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| Metadata: |    |
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| Categories: | Misc.
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