We study the approximate maximum weight matching (MWM) problem in a fully dynamic graph subject to edge insertions and deletions. We design meta-algorithms that reduce the problem to the unweighted approximate maximum cardinality matching (MCM) problem. Despite recent progress on bipartite graphs -- Bernstein-Dudeja-Langley (STOC 2021) and Bernstein-Chen-Dudeja-Langley-Sidford-Tu (SODA 2025) -- the only previous meta-algorithm that applied to non-bipartite graphs suffered a $\frac{1}{2}$ approximation loss (Stubbs-Williams, ITCS 2017). We significantly close the weighted-and-unweighted gap by showing the first low-loss reduction that transforms any fully dynamic $(1-\varepsilon)$-approximate MCM algorithm on bipartite graphs into a fully dynamic $(1-\varepsilon)$-approximate MWM algorithm on general (not necessarily bipartite) graphs, with only a $\mathrm{poly}(\log n/\varepsilon)$ overhead in the update time. Central to our approach is a new primal-dual framework that reduces the computation of an approximate MWM in general graphs to a sequence of approximate induced matching queries on an auxiliary bipartite extension. In addition, we give the first conditional lower bound on approximate partially dynamic matching with worst-case update time.

翻译：我们研究了在边插入和删除的全动态图中近似最大权重匹配问题。我们设计了元算法，将问题归约为无权近似最大基数匹配问题。尽管二分图上的研究近期取得了进展——Bernstein-Dudeja-Langley（STOC 2021）和Bernstein-Chen-Dudeja-Langley-Sidford-Tu（SODA 2025）——但此前唯一适用于非二分图的元算法存在$\frac{1}{2}$的近似损失（Stubbs-Williams，ITCS 2017）。我们通过提出首个低损失归约方法显著缩小了有权与无权问题之间的差距：该方法可将二分图上任何全动态$(1-\varepsilon)$-近似MCM算法转化为一般（非必二分）图上全动态$(1-\varepsilon)$-近似MWM算法，且更新时间仅产生$\mathrm{poly}(\log n/\varepsilon)$的开销。我们方法的核心是新的原始对偶框架，将一般图中近似MWM的计算归约为辅助二分扩展图上的一系列近似诱导匹配查询。此外，我们首次给出了最坏情况下更新时间条件下近似部分动态匹配问题的条件性下界。