We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a $(2-\delta)$-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a $(1+\delta)$-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintaining a $(2+\delta)$-approximate maximum matching in a fully dynamic general graph, in small polylogarithmic (specifically, $O(\log^4 n)$) update time. These results are respectively obtained by applying our framework on top of the fractional matching algorithms of Bhattacharya et al. [STOC'16], Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19]. Prior to our work, there were two known general-purpose rounding schemes for dynamic fractional matchings. Both these schemes, by Arar et al. [ICALP'18] and Wajc [STOC'20], were randomized. Our rounding scheme works by maintaining a good {\em matching-sparsifier} with bounded arboricity, and then applying the algorithm of Peleg and Solomon [SODA'16] to maintain a near-optimal matching in this low arboricity graph. To the best of our knowledge, this is the first dynamic matching algorithm that works on general graphs by using an algorithm for low-arboricity graphs as a black-box subroutine. This feature of our rounding scheme might be of independent interest.

翻译：我们展示了一个框架, 用于在任意的小型多面性更新时间里, 以确定性地四舍五入一个动态分数匹配。 在现有的分数匹配算法之外, 以黑箱方式应用我们的框架, 我们得出以下新结果:(1) 在一个完全动态的双面图中, 使用第一个确定性算法来保持$( 2-\ delta) $- 近似的最大匹配。 (2) 在任意的小型多面性更新时间里, 使用第一个确定性算法来保持一个 $( 1 ⁇ delta) 的圆形双向匹配。 在多面性双面性图中, 以黑面性双面图的形式, 在多面性更新的时间里, 以黑面性平面性平面性算法 。 [FOC'20] 和 Bhattharterminalalalalalalality 匹配算法, 以我们已知的双面性平面性平面性平面性平面性平面性平面性平面性平面图 。