Hitting Sets Online and Unique-Max Coloring

TL;DR

This paper investigates online hitting set problems for hypergraphs with a specific union property, establishing bounds on the competitive ratio based on the unique-max number.

Contribution

It introduces bounds on the competitive ratio for a class of hypergraphs where the union of intersecting ranges is also a range, linking it to the unique-max number.

Findings

Competitive ratio is at most the unique-max number.

Competitive ratio is at least the unique-max number minus one.

Results apply to hypergraphs with union-closed intersecting ranges.

Abstract

We consider the problem of hitting sets online. The hypergraph (i.e., range-space consisting of points and ranges) is known in advance, and the ranges to be stabbed are input one-by-one in an online fashion. The online algorithm must stab each range upon arrival. An online algorithm may add points to the hitting set but may not remove already chosen points. The goal is to use the smallest number of points. The best known competitive ratio for hitting sets online by Alon et al. \cite{alon2009online} is $O(\log n \cdot \log m)$ for general hypergraphs, where $n$ and $m$ denote the number of points and the number of ranges, respectively. We consider hypergraphs in which the union of two intersecting ranges is also a range. Our main result for such hypergraphs is as follows. The competitive ratio of the online hitting set problem is at most the unique-max number and at least this number minus one.