Dynamic Edge Coloring of Forests

TL;DR

This paper studies dynamic edge coloring of forests, proposing algorithms with optimal or near-optimal recourse bounds in incremental and fully dynamic models, including deterministic and randomized approaches.

Contribution

It introduces new algorithms for dynamic edge coloring of forests that optimize recourse, improving upon greedy methods and addressing both incremental and fully dynamic scenarios.

Findings

Greedy algorithm achieves tight $O(1/(c + \sqrt{\Delta}))$ recourse in incremental model.

Greedy can have $\Omega(\log_\Delta n)$ recourse in fully dynamic forests.

Non-greedy algorithm attains $O(1)$ recourse for rooted forests with $c=\Delta-2$.

Abstract

In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $\Delta$ with at most $\Delta+c$ 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 $O(\frac{1}{c + \sqrt{\Delta}})$ 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 $\Omega(\log_\Delta n)$ 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 $c = \Delta - 2$. In the randomized setting, we give a natural distribution-maintaining algorithm that achieves $\Theta(\frac{1}{\Delta})$ expected amortized recourse in the incremental model and $\Theta(\min \{ \frac{\Delta}{c}, \log_{\Delta} n \})$ expected recourse in the dynamic model. These randomized results are optimal for $c=0$.