Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems
This work provides improved algorithms for online hitting set in structured domains, offering incremental gains over general methods for problems in computational geometry and optimization.
The paper tackles the online hitting set problem for structured and geometric set systems by developing an algorithm with competitive ratio O(log n log log n), improving upon the previous O(log n log m) bound for general cases and addressing open questions in geometric settings.
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.