# Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

## TL;DR

This paper studies the problem of maintaining conflict-free colorings of intervals with respect to points in a dynamic setting, providing bounds on recoloring costs and number of colors, including for kinetic scenarios.

## Contribution

It introduces the fully-dynamic conflict-free coloring problem for intervals, establishes bounds on recoloring costs versus number of colors, and explores kinetic case solutions.

## Key findings

- Lower bound on recolorings related to number of colors
- Algorithms using O(log n) colors with O(log n) recolorings
- Kinetic setting with 4-color coloring and tight bounds

## Abstract

We introduce the fully-dynamic conflict-free coloring problem for a set $S$ of intervals in $\mathbb{R}^1$ with respect to points, where the goal is to maintain a conflict-free coloring for$S$ under insertions and deletions. A coloring is conflict-free if for each point $p$ contained in some interval, $p$ is contained in an interval whose color is not shared with any other interval containing $p$. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with $O(1)$ recolorings per update the worst-case number of colors is $\Omega(\log n/\log\log n)$, and that any strategy using $O(1/\varepsilon)$ colors needs $\Omega(\varepsilon n^{\varepsilon})$ recolorings; - a coloring strategy that uses $O(\log n)$ colors at the cost of $O(\log n)$ recolorings, and another strategy that uses $O(1/\varepsilon)$ colors at the cost of $O(n^{\varepsilon}/\varepsilon)$ recolorings; - stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.