局部边际的力量：动态加权匹配的 $O(\varepsilon^{-1})$ 纵横比归约
The Power of Local Marginals: An $O(\varepsilon^{-1})$-Aspect-Ratio Reduction for Dynamic Weighted Matching
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对动态最大权匹配问题，基于局部边际结构性质提出一种归约方法，将多项式纵横比实例转化为 $O(\varepsilon^{-1})$ 纵横比实例，改进了既有数值归约与显式匹配合成的局部纵横比界。
AI中文摘要：
我们在两种场景下研究边插入和删除操作下的动态最大权匹配（MWM）：维护最优权值的 $(1\pm\varepsilon)$ 近似解，以及维护显式的 $(1-\varepsilon)$ 近似匹配。我们的核心结果是一种归约方法，可将多项式纵横比的实例转化为纵横比为 $O(\varepsilon^{-1})$ 的实例。该归约适用于两种场景下的一般图，且与部分动态更新兼容。\n该归约基于局部边际的结构性质。将边按权重分组后，某一权重类相对于所有更低权重类的全局边际贡献，可通过其在纵横比为 $O(\varepsilon^{-1})$ 的局部权重窗口内的边际贡献近似。将这些局部边际求和，可得到仅需使用局部窗口近似最优值的数值合成引理。这改进了Gupta和Peng（FOCS 2013）的数值归约，其局部纵横比为 $\varepsilon^{-\Theta(\varepsilon^{-1})}$。相同的结构性质还为显式匹配带来了改进的匹配合成引理，将Bernstein--Chen--Dudeja--Langley--Sidford--Tu（SODA 2025）的局部纵横比从 $O(\varepsilon^{-2})$ 降至 $O(\varepsilon^{-1})$。
英文摘要：
We study dynamic maximum weight matching under edge insertions and deletions in two settings: maintaining a $(1\pm\varepsilon)$-approximation to the optimum weight, and maintaining an explicit $(1-\varepsilon)$-approximate matching. Our main result is a reduction that transforms instances of polynomial aspect ratio into instances of aspect ratio $O(\varepsilon^{-1})$. The reduction applies to general graphs in both settings and is compatible with partially dynamic updates. The reduction is based on a structural property of local marginals. After grouping edges into weight classes, the global marginal contribution of one class relative to all lower classes is approximated by its marginal contribution within a local weight window of aspect ratio $O(\varepsilon^{-1})$. Summing these local marginals yields a value composition lemma that approximates the optimum weight in the entire graph with approximate optimum weights of the local windows. This improves the value reduction of Gupta and Peng (FOCS 2013), whose local aspect ratio is $\varepsilon^{-Θ(\varepsilon^{-1})}$. The same structural property yields an improved matching composition lemma for explicit matchings, reducing the local aspect ratio of Bernstein--Chen--Dudeja--Langley--Sidford--Tu (SODA 2025) from $O(\varepsilon^{-2})$ to $O(\varepsilon^{-1})$.
