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# Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set

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LIPIcs.SoCG.2023.46.pdf
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## Cite As

Arindam Khan, Aditya Lonkar, Saladi Rahul, Aditya Subramanian, and Andreas Wiese. Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.46

## Abstract

Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), m sets (resp. n points) are given and n points (resp. m sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight O(log n)-competitive algorithm. In the same setting for hitting set, we provide a tight O(log N)-competitive algorithm, assuming that all points have integral coordinates in [0,N)². No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems). For both dynamic set cover and hitting set with d-dimensional hyperrectangles, we obtain (log m)^O(d)-approximation algorithms with (log m)^O(d) worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an extended quad-tree approach and a frequency reduction technique that reduces geometric set cover instances to instances of general set cover with bounded frequency.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Theory of computation → Online algorithms
##### Keywords
• Geometric Set Cover
• Hitting Set
• Rectangles
• Squares
• Hyperrectangles
• Online Algorithms
• Dynamic Data Structures

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## References

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