On Stable Approximation Algorithms for Geometric Coverage Problems
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915069399400448 |
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| author | de Berg, Mark Sadhukhan, Arpan |
| author_facet | de Berg, Mark Sadhukhan, Arpan |
| contents | Let $P$ be a set of points in the plane and let $m$ be an integer. The goal of Max Cover by Unit Disks problem is to place $m$ unit disks whose union covers the maximum number of points from~$P$. We are interested in the dynamic version of Max Cover by Unit Disks problem, where the points in $P$ appear and disappear over time, and the algorithm must maintain a set \cDalg of $m$ disks whose union covers many points. A dynamic algorithm for this problem is a $k$-stable $α$-approximation algorithm when it makes at most $k$ changes to \cDalg upon each update to the set $P$ and the number of covered points at time $t$ is always at least $α\cdot \opt(t)$, where $\opt(t)$ is the maximum number of points that can be covered by m disks at time $t$. We show that for any constant $\varepsilon>0$, there is a $k_{\varepsilon}$-stable $(1-\varepsilon)$-approximation algorithm for the dynamic Max Cover by Unit Disks problem, where $k_{\varepsilon}=O(1/\varepsilon^3)$. This improves the stability of $Θ(1/\eps^4)$ that can be obtained by combining results of Chaplick, De, Ravsky, and Spoerhase (ESA 2018) and De~Berg, Sadhukhan, and Spieksma (APPROX 2023). Our result extends to other fat similarly-sized objects used in the covering, such as arbitrarily-oriented unit squares, or arbitrarily-oriented fat ellipses of fixed diameter.
We complement the above result by showing that the restriction to fat objects is necessary to obtain a SAS. To this end, we study the Max Cover by Unit Segments problem, where the goal is to place $m$ unit-length segments whose union covers the maximum number of points from $P$. We show that there is a constant $\varepsilon^* > 0$ such that any $k$-stable $(1 + \varepsilon^*)$-approximation algorithm must have $k=Ω(m)$, even when the point set never has more than four collinear points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Stable Approximation Algorithms for Geometric Coverage Problems de Berg, Mark Sadhukhan, Arpan Computational Geometry Let $P$ be a set of points in the plane and let $m$ be an integer. The goal of Max Cover by Unit Disks problem is to place $m$ unit disks whose union covers the maximum number of points from~$P$. We are interested in the dynamic version of Max Cover by Unit Disks problem, where the points in $P$ appear and disappear over time, and the algorithm must maintain a set \cDalg of $m$ disks whose union covers many points. A dynamic algorithm for this problem is a $k$-stable $α$-approximation algorithm when it makes at most $k$ changes to \cDalg upon each update to the set $P$ and the number of covered points at time $t$ is always at least $α\cdot \opt(t)$, where $\opt(t)$ is the maximum number of points that can be covered by m disks at time $t$. We show that for any constant $\varepsilon>0$, there is a $k_{\varepsilon}$-stable $(1-\varepsilon)$-approximation algorithm for the dynamic Max Cover by Unit Disks problem, where $k_{\varepsilon}=O(1/\varepsilon^3)$. This improves the stability of $Θ(1/\eps^4)$ that can be obtained by combining results of Chaplick, De, Ravsky, and Spoerhase (ESA 2018) and De~Berg, Sadhukhan, and Spieksma (APPROX 2023). Our result extends to other fat similarly-sized objects used in the covering, such as arbitrarily-oriented unit squares, or arbitrarily-oriented fat ellipses of fixed diameter. We complement the above result by showing that the restriction to fat objects is necessary to obtain a SAS. To this end, we study the Max Cover by Unit Segments problem, where the goal is to place $m$ unit-length segments whose union covers the maximum number of points from $P$. We show that there is a constant $\varepsilon^* > 0$ such that any $k$-stable $(1 + \varepsilon^*)$-approximation algorithm must have $k=Ω(m)$, even when the point set never has more than four collinear points. |
| title | On Stable Approximation Algorithms for Geometric Coverage Problems |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2412.13357 |