🤖 AI Summary
This paper studies the online hitting set problem: unit balls or unit hypercubes in ℝ^s arrive sequentially, and an online algorithm must select a minimum-size hitting set from the integer lattice ℤ^d in real time. The core challenge lies in worst-case competitive ratio analysis under adversarial, unknown geometric object sequences. We establish the first tight competitive ratio bounds: Ω(2^s) lower bound and O(2^s) upper bound—demonstrating that the performance bottleneck is governed solely by the ambient space dimension s, independent of the lattice dimension d. Our approach integrates combinatorial geometry, discrete covering theory, and online algorithm analysis to reveal an optimal strategy structure under dimensional separation: the s-dimensional geometric complexity fundamentally dictates the resource-efficiency limit. This work provides the first tight competitive ratio characterization for online hitting sets against both unit balls and unit hypercubes—two fundamental geometric families.
