Online Geometric Hitting Set and Set Cover Beyond Unit Balls in ℝ^2
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We investigate the geometric hitting set problem in the online setup for the range space Σ=( P, S), where the set ⊂ℝ^2 is a collection of n points and the set S is a family of geometric objects in 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 Σ is known in advance, but the order of arrival of the input objects in S is unknown. They proposed online algorithms having optimal competitive ratios of Θ(log n) for intervals, half-planes and unit disks in 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 Σ is not known in advance. We only know the set P but not the set 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 Θ(log(n)) for unit disks and regular k-gon with k≥ 4 in ℝ^2. All the above-mentioned results also hold for the equivalent geometric set cover problem in Model-II.
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