Online Hitting Set for Axis-Aligned Squares

TL;DR

This paper introduces an optimal logarithmic-competitive deterministic algorithm for the online hitting set problem with axis-aligned squares, extending to homothets of polygons with multiple vertices, advancing geometric online algorithms.

Contribution

It presents the first $O( ext{log } n)$-competitive algorithm for arbitrary-sized geometric objects in the plane, generalizing to polygons with $k$ vertices.

Findings

Achieves an $O( ext{log } n)$ competitive ratio for axis-aligned squares.

Extends the approach to positive homothets of polygons with $k ext{ vertices}$.

Provides the first such optimal ratio for geometric objects of arbitrary sizes.

Abstract

We are given a set $P$ of $n$ points in the plane, and a sequence of axis-aligned squares that arrive in an online fashion. The online hitting set problem consists of maintaining, by adding new points if necessary, a set $H\subseteq P$ that contains at least one point in each input square. We present an $O(\log n)$-competitive deterministic algorithm for this problem. The competitive ratio is the best possible, apart from constant factors. In fact, this is the first $O(\log n)$-competitive algorithm for the online hitting set problem that works for geometric objects of arbitrary sizes (i.e., arbitrary scaling factors) in the plane. We further generalize this result to positive homothets of a polygon with $k\geq 3$ vertices in the plane and provide an $O(k^2\log n)$-competitive algorithm.