Dynamic Edge Coloring of Forests

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 $c=0$.