Recoloring Unit Interval Graphs with Logarithmic Recourse Budget
Bart{\l}omiej Bosek, Anna Zych-Pawlewicz

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 -colorable at all times, and the updates consist of insertions only, then we can achieve the amortized recourse budget of while maintaining a proper coloring with 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 and . We complement this result by showing the lower bound of…
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