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### Online Piercing of Geometric Objects

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### Abstract

We consider the online version of the piercing set problem where geometric objects arrive one by one. The online algorithm must maintain a piercing set for the arrived objects by making irrevocable decisions. First, we show that any deterministic online algorithm that solves this problem has a competitive ratio of at least Ω(n), which even holds when the objects are one-dimensional intervals. On the other hand, piercing unit objects is equivalent to the unit covering problem which is well-studied in the online model. Due to this, all the results related to the online unit covering problem are preserved for the online unit piercing problem when the objects are translated from each other. Surprisingly, no upper bound was known for the unit covering problem when unit objects are anything other than balls and hypercubes. In this paper, we introduce the notion of α-aspect and α-aspect_∞ objects. We give an upper bound of competitive ratio for α-aspect and α-aspect_∞ objects in ℝ³ and ℝ^d, respectively, with a scaling factor in the range [1,k]. We also propose a lower bound of the competitive ratio for bounded scaled objects like α-aspect objects in ℝ², axis-aligned hypercubes in ℝ^d, and balls in ℝ² and ℝ³. For piercing α-aspect_∞ objects in ℝ^d, we show that a simple deterministic algorithm achieves a competitive ratio of at most (2/α)^d((1+α)^d-1) (⌈log_(1+α)(2k/α)⌉)+1. This result is very general in nature. One can obtain upper bounds for specific objects by specifying the value of α. By putting the value of k = 1 to the above result, we get an upper bound of the competitive ratio for the unit covering problem for various types of objects.

### BibTeX - Entry

```@InProceedings{de_et_al:LIPIcs.FSTTCS.2022.17,
author =	{De, Minati and Jain, Saksham and Kallepalli, Sarat Varma and Singh, Satyam},
title =	{{Online Piercing of Geometric Objects}},
booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages =	{17:1--17:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-261-7},
ISSN =	{1868-8969},
year =	{2022},
volume =	{250},
editor =	{Dawar, Anuj and Guruswami, Venkatesan},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},