Given a graph $G$ that is modified by a sequence of edge insertions and deletions, we study the Maximum $k$-Edge Coloring problem Having access to $k$ colors, how can we color as many edges of $G$ as possible such that no two adjacent edges share the same color? While this problem is different from simply maintaining a $b$-matching with $b=k$, the two problems are closely related: a maximum $k$-matching always contains a $\frac{k+1}k$-approximate maximum $k$-edge coloring. However, maximum $b$-matching can be solved efficiently in the static setting, whereas the Maximum $k$-Edge Coloring problem is NP-hard and even APX-hard for $k \ge 2$. We present new results on both problems: For $b$-matching, we show a new integrality gap result and for the case where $b$ is a constant, we adapt Wajc's matching sparsification scheme~[STOC20]. Using these as basis, we give three new algorithms for the dynamic Maximum $k$-Edge Coloring problem: Our MatchO algorithm builds on the dynamic $(2+\epsilon)$-approximation algorithm of Bhattacharya, Gupta, and Mohan~[ESA17] for $b$-matching and achieves a $(2+\epsilon)\frac{k+1} k$-approximation in $O(poly(\log n, \epsilon^{-1}))$ update time against an oblivious adversary. Our MatchA algorithm builds on the dynamic $8$-approximation algorithm by Bhattacharya, Henzinger, and Italiano~[SODA15] for fractional $b$-matching and achieves a $(8+\epsilon)\frac{3k+3}{3k-1}$-approximation in $O(poly(\log n, \epsilon^{-1}))$ update time against an adaptive adversary. Moreover, our reductions use the dynamic $b$-matching algorithm as a black box, so any future improvement in the approximation ratio for dynamic $b$-matching will automatically translate into a better approximation ratio for our algorithms. Finally, we present a greedy algorithm that runs in $O(\Delta+k)$ update time, while guaranteeing a $2.16$~approximation factor.

翻译：给定一个通过一系列边插入和删除操作修改的图$G$，我们研究最大$k$-边着色问题：在拥有$k$种颜色的条件下，如何为$G$中尽可能多的边着色，使得任意两条相邻边颜色不同？虽然该问题与维护一个$b=k$的$b$-匹配不同，但两者密切相关：最大$k$-匹配始终包含一个$\frac{k+1}{k}$近似度的最大$k$-边着色。然而，最大$b$-匹配在静态场景下可高效求解，而最大$k$-边着色问题是NP难的，且对于$k \ge 2$甚至是APX难的。我们针对这两个问题提出了新结果：对于$b$-匹配，我们展示了新的整数性间隙结果；当$b$为常数时，我们改进了Wajc的匹配稀疏化方案[STOC20]。以此为基础，我们为动态最大$k$-边着色问题提出了三种新算法：我们的MatchO算法基于Bhattacharya、Gupta和Mohan [ESA17]的动态$(2+\epsilon)$近似$b$-匹配算法，在对抗性半诚实对手下实现了$O(poly(\log n, \epsilon^{-1}))$更新时间和$(2+\epsilon)\frac{k+1}{k}$近似比。我们的MatchA算法基于Bhattacharya、Henzinger和Italiano [SODA15]的分数$b$-匹配动态$8$近似算法，在自适应对手下实现了$O(poly(\log n, \epsilon^{-1}))$更新时间和$(8+\epsilon)\frac{3k+3}{3k-1}$近似比。此外，我们的归约过程将动态$b$-匹配算法视为黑箱，因此未来任何动态$b$-匹配近似比的改进都将自动转化为我们算法近似比的提升。最后，我们提出了一种贪心算法，其更新时间为$O(\Delta+k)$，同时保证$2.16$的近似因子。