Computing largest minimum color-spanning intervals of imprecise points
<|reference_start|>Computing largest minimum color-spanning intervals of imprecise points: We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning interval} is as large as possible. A minimum color-spanning interval is an interval of minimum size that contains at least one point from a given interval of each color. We prove that if the input intervals are pairwise disjoint, the problem can be solved in $O(n)$ time, even for intervals of arbitrary length. For overlapping intervals, the problem becomes much more difficult. Nevertheless, we show that it can be solved in $O(n \log^2 n)$ time when $k=2$, by exploiting several structural properties of candidate solutions, combined with a number of advanced algorithmic techniques. Interestingly, this shows a sharp contrast with the 2-dimensional version of the problem, recently shown to be NP-hard.<|reference_end|>
@article{acharyya2024computing,
    title={Computing largest minimum color-spanning intervals of imprecise points},
    author={Ankush Acharyya, Vahideh Keikha, Maria Saumell, Rodrigo I. Silveira},
    journal={arXiv preprint arXiv:2410.03213},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.03213},
    primaryClass={cs.CG cs.DS}
}