We study dynamic $(1-\epsilon)$-approximate rounding of fractional matchings -- a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time $O(\epsilon^{-1} \log^2 (\epsilon^{-1} \cdot n))$, matching an (unconditional) recourse lower bound of $\Omega(\epsilon^{-1})$ up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic \emph{partial rounding} algorithms to increase this minimum weight, we obtain several algorithms that improve this dependence on $n$. For example, we give a high-probability randomized algorithm with $\tilde{O}(\epsilon^{-1}\cdot (\log\log n)^2)$-update time against adaptive adversaries. (We use Soft-Oh notation, $\tilde{O}$, to suppress polylogarithmic factors in the argument, i.e., $\tilde{O}(f)=O(f\cdot \mathrm{poly}(\log f))$.) Using our rounding algorithms, we also round known $(1-\epsilon)$-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.

翻译：我们研究分数匹配的动态$(1-\epsilon)$-近似舍入问题——这是动态图算法领域诸多突破性成果中的关键组成部分。本文首先提出一个极其简洁的确定性二分图舍入算法，其平摊更新时间达到$O(\epsilon^{-1} \log^2 (\epsilon^{-1} \cdot n))$，与无条件在线性对数因子范围内的下界$\Omega(\epsilon^{-1})$相匹配。此外，当分数匹配中最小（非零）权重在整个过程中保持有下界时，该算法的更新时间可进一步改善。通过将该算法与新型动态\emph{部分舍入}算法（用于提升最小权重）相结合，我们获得多个改进对$n$依赖关系的算法。例如，我们给出一个针对自适应对手具有$\tilde{O}(\epsilon^{-1}\cdot (\log\log n)^2)$更新时间的高概率随机算法。（本文使用软O符号$\tilde{O}$表示忽略参数中的多对数因子，即$\tilde{O}(f)=O(f\cdot \mathrm{poly}(\log f))$。）基于我们的舍入算法，我们还能在无渐近开销的情况下对已知的$(1-\epsilon)$-递减二分图分数匹配算法进行舍入，从而改进了递减二分图匹配问题的现有最优算法。此外，我们将结果扩展至一般图并实现了近似最大匹配的维护。