Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems

TL;DR

This paper presents an improved online hitting set algorithm with a better competitive ratio for structured and geometric set systems, especially those with linear shallow-cell complexity, advancing the understanding of online algorithms in geometric contexts.

Contribution

It introduces an $O(\log n \log\log n)$-competitive algorithm for weighted online hitting set on set systems with linear shallow-cell complexity, surpassing previous bounds.

Findings

Achieved a single-logarithmic competitive ratio for certain geometric set systems.

First bounds established for weighted online hitting set in natural geometric families.

Replaced the double-logarithmic barrier in general set systems with a more efficient ratio.

Abstract

In the online hitting set problem, sets arrive over time, and the algorithm has to maintain a subset of elements that hit all the sets seen so far. Alon, Awerbuch, Azar, Buchbinder, and Naor (SICOMP 2009) gave an algorithm with competitive ratio $O(\log n \log m)$ for the (general) online hitting set and set cover problems for $m$ sets and $n$ elements; this is known to be tight for efficient online algorithms. Given this barrier for general set systems, we ask: can we break this double-logarithmic phenomenon for online hitting set/set cover on structured and geometric set systems? We provide an $O(\log n \log\log n)$-competitive algorithm for the weighted online hitting set problem on set systems with linear shallow-cell complexity, replacing the double-logarithmic factor in the general result by effectively a single logarithmic term. As a consequence of our results we obtain the first bounds for weighted online hitting set for natural geometric set families, thereby answering open questions regarding the gap between general and geometric weighted online hitting set problems.