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- W4366196672 abstract "We investigate the geometric hitting set problem in the online setup for the range space $Sigma=({cal P},{cal S})$, where the set $Psubsetmathbb{R}^2$ is a collection of $n$ points and the set $cal S$ is a family of geometric objects in $mathbb{R}^2$. In the online setting, the geometric objects arrive one by one. Upon the arrival of an object, an online algorithm must maintain a valid hitting set by making an irreversible decision, i.e., once a point is added to the hitting set by the algorithm, it can not be deleted in the future. The objective of the geometric hitting set problem is to find a hitting set of the minimum cardinality. Even and Smorodinsky (Discret. Appl. Math., 2014) considered an online model (Model-I) in which the range space $Sigma$ is known in advance, but the order of arrival of the input objects in $cal S$ is unknown. They proposed online algorithms having optimal competitive ratios of $Theta(log n)$ for intervals, half-planes and unit disks in $mathbb{R}^2$. Whether such an algorithm exists for unit squares remained open for a long time. This paper considers an online model (Model-II) in which the entire range space $Sigma$ is not known in advance. We only know the set $cal P$ but not the set $cal S$ in advance. Note that any algorithm for Model-II will also work for Model-I, but not vice-versa. In Model-II, we obtain an optimal competitive ratio of $Theta(log(n))$ for unit disks and regular $k$-gon with $kgeq 4$ in $mathbb{R}^2$. All the above-mentioned results also hold for the equivalent geometric set cover problem in Model-II." @default.
- W4366196672 created "2023-04-19" @default.
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- W4366196672 date "2023-04-13" @default.
- W4366196672 modified "2023-10-16" @default.
- W4366196672 title "Online Geometric Hitting Set and Set Cover Beyond Unit Balls in $mathbb{R}^2$" @default.
- W4366196672 doi "https://doi.org/10.48550/arxiv.2304.06780" @default.
- W4366196672 hasPublicationYear "2023" @default.
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