Recoloring Unit Interval Graphs with Logarithmic Recourse Budget

TL;DR

This paper presents an efficient incremental coloring algorithm for dynamic unit interval graphs with a logarithmic recourse budget, significantly improving previous bounds and providing new insights into circular arc graph coloring.

Contribution

The paper introduces a new incremental coloring algorithm for k-colorable unit interval graphs with logarithmic recourse, and establishes a lower bound on recourse in the dynamic setting.

Findings

Achieves amortized recourse of O(k^7 log n) for insert-only updates.

Provides a lower bound of Ω(n) on recourse in the fully dynamic setting.

Includes a new result on coloring proper circular arc graphs based on intersection properties.

Abstract

In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals is allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains $k$-colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of $O(k^7 \log n)$ while maintaining a proper coloring with $k$ colors. This is an exponential improvement over the result in [Bosek et al., Recoloring Interval Graphs with Limited Recourse Budget. SWAT 2020] in terms of both $k$ and $n$. We complement this result by showing the lower bound of $\Omega(n)$ on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As a byproduct of independent interest we include a new result on coloring proper circular arc graphs. Let $L$ be the maximum number of arcs intersecting in one point for some set of unit circular arcs $\mathcal{A}$. We show that if there is a set $\mathcal{A}'$ of non-intersecting unit arcs of size $L^2-1$ such that $\mathcal{A} \cup \mathcal{A}'$ does not contain $L+1$ arcs intersecting in one point, then it is possible to color $\mathcal{A}$ with $L$ colors. This complements the work on unit circular arc coloring, which specifies sufficient conditions needed to color $\mathcal{A}$ with $L+1$ colors or more.