We consider the foundational problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching (MWM) in an $n$-node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem on graphs with a weight range of $\mathrm{poly}(n)$ to $\mathrm{poly}(1/\varepsilon)$ at the cost of just an additive $\mathrm{poly}(1/\varepsilon)$ in update time. This improves upon the prior reduction of Gupta-Peng (FOCS 2013) which reduces the problem to a weight range of $\varepsilon^{-O(1/\varepsilon)}$ with a multiplicative cost of $O(\log n)$. When combined with a reduction of Bernstein-Dudeja-Langley (STOC 2021) this yields a reduction from dynamic $(1-\varepsilon)$-approximate MWM in bipartite graphs with a weight range of $\mathrm{poly}(n)$ to dynamic $(1-\varepsilon)$-approximate maximum cardinality matching in bipartite graphs at the cost of a multiplicative $\mathrm{poly}(1/\varepsilon)$ in update time, thereby resolving an open problem in [GP'13; BDL'21]. Additionally, we show that our approach is amenable to MWM problems in streaming, shared-memory work-depth, and massively parallel computation models. We also apply our techniques to obtain an efficient dynamic algorithm for rounding weighted fractional matchings in general graphs. Underlying our framework is a new structural result about MWM that we call the "matching composition lemma" and new dynamic matching subroutines that may be of independent interest.

翻译：我们研究在一个经历边插入与删除的 $n$ 节点动态图中，维护一个 $(1-\varepsilon)$-近似最大权重匹配（MWM）的基础性问题。我们提出了一种通用约简方法，将权重范围为 $\mathrm{poly}(n)$ 的图上的该问题，约简到权重范围为 $\mathrm{poly}(1/\varepsilon)$ 的问题，而更新时间的代价仅为 $\mathrm{poly}(1/\varepsilon)$ 的加法项。这改进了 Gupta-Peng（FOCS 2013）先前提出的约简方法，后者将问题约简到权重范围为 $\varepsilon^{-O(1/\varepsilon)}$ 的情况，并付出了 $O(\log n)$ 的乘法代价。当与 Bernstein-Dudeja-Langley（STOC 2021）的约简方法结合时，我们的方法实现了从权重范围为 $\mathrm{poly}(n)$ 的二部图中的动态 $(1-\varepsilon)$-近似 MWM 问题，到二部图中动态 $(1-\varepsilon)$-近似最大基数匹配问题的约简，其更新时间的代价为 $\mathrm{poly}(1/\varepsilon)$ 的乘法项，从而解决了 [GP'13; BDL'21] 中提出的一个开放性问题。此外，我们证明了我们的方法适用于流式计算、共享内存工作深度模型以及大规模并行计算模型中的 MWM 问题。我们还应用我们的技术，为一般图中加权分数匹配的舍入问题提出了一种高效的动态算法。我们框架的基础是一个关于 MWM 的新结构结果，我们称之为"匹配组合引理"，以及一些可能具有独立研究价值的新的动态匹配子程序。