Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems

2026-03-15

AI Summary

This work addresses the online hitting set problem, where the goal is to maintain a subset of elements that intersects all sets as they arrive one by one. Focusing on geometric set systems with linear shallow-cell complexity, the paper presents the first weighted online algorithm that improves the competitive ratio from the general lower bound of $O(\log n \log m)$ to $O(\log n \log \log n)$. This advance is achieved by integrating shallow-cell complexity analysis, the potential function method, and online primal-dual fitting techniques. The result establishes the first near-single-logarithmic competitive ratio for canonical geometric settings, providing the first theoretical guarantees for several natural families of geometric sets and bridging the gap between general and structured instances in online hitting set theory.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Heuristic Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• online hitting set
• geometric set systems
• competitive ratio
• structured set systems
• shallow-cell complexity

Innovation

• online hitting set
• shallow-cell complexity
• geometric set systems
• competitive ratio
• weighted set cover