Dynamic Edge Coloring of Forests
In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $\Delta$ with at most $\Delta+c$ colors, under edge updates. A prominent objective is to minimize the \emph{recourse}, namely the number of edges that are recolored. We study this problem on forests, arguably the simplest graph class that already captures much of the complexity of the problem. We consider both the \emph{incremental} model, where edges are only inserted and the \emph{fully dynamic} model where edges may also be deleted. In the deterministic setting, we focus on the natural greedy algorithm. We show that it achieves $O(\frac{1}{c + \sqrt{\Delta}})$ amortized recourse in the incremental model, and that 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 overcome this limitation of greedy within the deterministic setting, we give 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. In the incremental model, it achieves $\Theta(\frac{1}{\Delta})$ expected amortized recourse, and we show that this is optimal for every constant $c$. In the fully dynamic model, the same algorithm achieves $\Theta(\min \{ \frac{\Delta}{c}, \log_{\Delta} n \})$ expected recourse for $c > 0$, and $\Theta(\log_{\Delta} n)$ for $c = 0$. We show that this is optimal for $c = 0$, and prove an $\Omega(1)$ lower bound for every constant $c$.