Dynamic Edge Coloring of Forests

2026-05-10

AI Summary

This work addresses the edge-coloring problem on dynamic forests, aiming to minimize the number of recolored edges (recourse) using at most Δ + c colors. For both incremental and fully dynamic settings, the paper introduces deterministic greedy and non-greedy reconstruction strategies, as well as randomized color distribution maintenance techniques, analyzed via amortized and probabilistic methods. The main contributions include the first tight bound for the greedy algorithm in the incremental model, achieving O(1/(c + √Δ)) amortized recourse; the first O(1) amortized recourse non-greedy algorithm for rooted fully dynamic forests; and a randomized algorithm that attains optimal expected recourse of Θ(1/Δ) when c = 0 and Θ(min{Δ/c, log_Δ n}) otherwise.

Abstract

In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $Δ$ with at most $Δ+c$ colors, given updates to the edges of the graph. An important objective is to minimize the \emph{recourse}, which is the number of edges being recolored. We study this problem on forests, which is a natural yet nontrivial restriction of the problem. We consider the problem in both \emph{incremental} (edges are only inserted) and \emph{fully dynamic} (edges may be deleted) models. In the deterministic setting, we show that the natural greedy algorithm achieves $O(\frac{1}{c + \sqrtΔ})$ amortized recourse in the incremental model, and this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $Ω(\log_Δn)$ amortized recourse. To partially alleviate this limitation of greedy, we show an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c = Δ- 2$. In the randomized setting, we give a natural distribution-maintaining algorithm that achieves $Θ(\frac{1}Δ)$ expected amortized recourse in the incremental model and $Θ(\min \{ \fracΔ{c}, \log_Δ n \})$ expected recourse in the dynamic model. These randomized results are optimal for $c=0$.

Problem

• dynamic edge coloring
• forests
• recourse
• incremental
• fully dynamic

Innovation

• dynamic edge coloring
• recourse
• forests
• amortized analysis
• randomized algorithm