On the FirstFit Algorithm for Online Unit-Interval Coloring

TL;DR

This work analyzes the online unit-length interval coloring problem when intervals can be open or closed, focusing on the FirstFit algorithm. It introduces a generalized pivot-bound counting framework that extends the classic neighborhood bound to handle interactions via multiple pivot intervals and a structured neighborhood decomposition. The authors prove a tight $2\omega$ bound for integral endpoints and a global bound of $\lceil\frac{7}{3}\omega\rceil-2$ for arbitrary endpoints, advancing understanding of FirstFit on bounded-length inputs. The methodology provides a robust counting technique that may extend to other bounded-length interval families and sheds light on the impact of endpoint uncertainty on online coloring performance.

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 $2ω$ colors when all intervals have integral endpoints and an upper bound of $\lceil\frac{7}{3}ω\rceil-2$ colors for the general case, where $ω$ is the optimal number of colors needed for the input set of intervals.

Figures (5)

• Figure 1: Rows denote the color assigned by OPT, where actual colors denote the color assigned by FirstFit, where the order is based on the index.
• Figure 2: All possible positions of intervals intersecting $I$ per type of row. The intervals in $\mathcal{A}(I)$ are drawn in orange, green and blue, the intervals in $\mathcal{T}(I)$ in purple.
• Figure 3: All possible intersections with interval $D^\texttt{A}(I)$ per type of row. The intervals from $\mathcal{A}(I)$ that intersect with $D^\texttt{A}(I)$ are drawn in orange, the intervals in $\mathcal{T}(I)$ in purple and the remaining intervals intersecting $D^\texttt{A}(I)$ are drawn in yellow.
• Figure 4: An exemplary non-exhaustive list of possible rows per type of row. Yellow intervals intersect only interval $D^\texttt{M}(I)$, purple intervals intersect only $I$ and orange intervals intersect both $D^\texttt{M}(I)$ and $I$.
• Figure 5: An exemplary non-exhaustive list of possible rows per type of $\mathcal{R}_\texttt{2}(I)$. Yellow intervals intersect only interval $D^\texttt{AM}(I)$ (resp. $D^\texttt{A}(I)$), purple intervals intersect only $I$ and orange intervals intersect both $D^\texttt{AM}(I)$ (resp. $D^\texttt{A}(I)$) and $I$.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Definition 5
• Lemma 6
• Lemma 8
• Corollary 9
• Theorem 10
• Definition 11