Online Hitting Set for Axis-Aligned Squares

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.