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arXiv · 2410.03213

Computing largest minimum color-spanning intervals of imprecise points

Ankush Acharyya·Vahideh Keikha·Maria Saumell·Rodrigo I. Silveira
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Abstract

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.

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Ankush Acharyya, Vahideh Keikha, Maria Saumell, Rodrigo I. Silveira. 2024-10-04. Computing largest minimum color-spanning intervals of imprecise points. https://arxiv.org/abs/2410.03213

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