Online Geometric Covering and Piercing
Minati De, Saksham Jain, Sarat Varma Kallepalli, Satyam Singh

TL;DR
This paper introduces a deterministic online algorithm for the geometric piercing set problem for similarly sized fat objects in Euclidean space, providing bounds on competitive ratios and establishing lower bounds for various shapes.
Contribution
It presents the first deterministic online algorithm with competitive ratio bounds for piercing similarly sized convex objects in higher dimensions.
Findings
Deterministic algorithm for hypercubes with ratio at most $3^d\lceil ext{log}_2 k ceil + 2^d$.
Lower bounds on competitive ratios for $ ext{α}$-fat objects in 2D and hypercubes in higher dimensions.
Upper bound on the competitive ratio for the online unit covering problem with convex objects.
Abstract
We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in has a competitive ratio of at least . This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in . For homothetic hypercubes in with side length in the range , we propose a deterministic algorithm having a competitive ratio of at most~. In the end, we show deterministic lower bounds of the competitive ratio for similarly sized -fat objects in and homothetic hypercubes…
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Taxonomy
TopicsOptimization and Search Problems · Complexity and Algorithms in Graphs · Computational Geometry and Mesh Generation
