Computing largest minimum color-spanning intervals of imprecise points

TL;DR

It is proved that if the input intervals are pairwise disjoint, the problem can be solved in O(n) time, even for intervals of arbitrary length, by exploiting several structural properties of candidate solutions, combined with a number of advanced algorithmic techniques.

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.

Figures (8)

• Figure 1: Example of unit intervals with $k=3$ colors, with an optimal realization that results in four L-MCSI s (indicated in gray). Note that the three leftmost representatives also form a color-spanning interval, which is not minimum.
• Figure 2: An example where the chains of mCSI s are $\{\gamma_1, \gamma_4, \gamma_6\}$, $\{\gamma_2,\gamma_5\}$, $\{\gamma_3\}$.
• Figure 3: Illustration for the proof of Lemma \ref{['lem:alt-middle-q']}.
• Figure 4: A tabular solution ${\cal{T}}_i$. We note ${\cal{T}}_i$ is maximal and illustrates case (i) in Observation \ref{['obs:end-Mi']}.
• Figure 5: Two leftmost solutions for the same input.

Theorems & Definitions (59)

• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• lemma thmcounterlemma
• proof