On the FirstFit Algorithm for Online Unit-Interval Coloring

TL;DR

This paper analyzes the FirstFit online coloring algorithm for unit-intervals, introducing a new counting method to bound its performance and providing tight bounds for specific cases.

Contribution

It develops a generalized neighborhood bound and critical interval concept to analyze FirstFit's performance, achieving tight bounds for integral and general cases.

Findings

Tight bound of 2ω colors for integral endpoints.

Upper bound of (7/3)ω - 2 colors for the general case.

New counting method improves analysis accuracy.

Abstract

In this paper, we study the performance of the FirstFit algorithm for the online unit-length intervals coloring problem where the intervals can be either open or closed, which serves a further investigation towards the actual performance of FirstFit. We develop a sophisticated counting method by generalizing the classic neighborhood bound, which limits the color FirstFit can assign an interval by counting the potential intersections. In the generalization, we show that for any interval, there is a critical interval intersecting it that can help reduce the overestimation of the number of intersections, and it further helps bound the color an interval can be assigned. The technical challenge then falls on identifying these critical intervals that guarantee the effectiveness of counting. Using this new mechanism for bounding the color that FirstFit can assign an interval, we provide a tight analysis of $2\omega$ colors when all intervals have integral endpoints and an upper bound of $\lceil\frac{7}{3}\omega\rceil-2$ colors for the general case, where $\omega$ is the optimal number of colors needed for the input set of intervals.