Hitting Geometric Objects Online via Points in $$\mathbb {Z}^d$$

Abstract

AbstractIn this paper, we consider the online version of the minimum hitting set problem, where geometric objects arrive one by one. The online algorithm must maintain a hitting set for the arrived objects by making irrevocable decisions. Here the centers of objects and hitting points are in \(\mathbb {R}^d\) and \(\mathbb {Z}^d\), respectively. First, we show that for hitting unit intervals using points from \(\mathbb {Z}\), one can achieve a tight bound of 2. Then, we consider the case of d-dimensional unit hypercubes. At first, we prove that every deterministic online algorithm for hitting unit hypercubes has a competitive ratio of at least \(d+1\) when \(d\in \mathbb {N}\). Later, for \(d=2\), we propose a deterministic online algorithm with a competitive ratio of at most 4 and for \(d>2\), we propose a randomized algorithm having a competitive ratio of \(O(d^2)\). Next, we consider the case of unit balls in \(\mathbb {R}^d\). We show that every deterministic online algorithm for hitting balls has a competitive ratio of at least \(d+1\) when \(d<4\). Then, we propose a deterministic algorithm having a competitive ratio of at most \(O(d^4)\) and O(1), for \(d\ge 4\) and \(d=3\), respectively. At last, for unit disks in \(\mathbb {R}^2\), we propose a simple deterministic algorithm that achieves a competitive ratio of at most 4.KeywordsHitting setOnline algorithmCompetitive ratioUnit coveringGeometric objects