Computing largest minimum color-spanning intervals of imprecise points

TL;DR

This paper investigates a geometric facility location problem with imprecise data, providing efficient algorithms for disjoint intervals and a near-optimal solution for overlapping intervals when two colors are involved.

Contribution

It introduces the first linear-time algorithm for disjoint intervals and an $O(n \, \log^2 n)$ algorithm for overlapping intervals with two colors, contrasting with the NP-hardness in higher dimensions.

Findings

Disjoint intervals can be solved in $O(n)$ time.

Overlapping intervals with two colors can be solved in $O(n \log^2 n)$ time.

The problem is NP-hard in 2D, but efficiently solvable in 1D for certain cases.

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.