We investigate dynamic maximum‑weight matching (MWM) under edge insertions and deletions, focusing on two goals: maintaining a $(1\pm\varepsilon)$ approximation to the optimal weight, and maintaining an explicit $(1-\varepsilon)$‑approximate matching.
The main contribution is a reduction that transforms instances with polynomial aspect ratio into instances whose aspect ratio is only $O(\varepsilon^{-1})$. The reduction works for general graphs in both settings and is compatible with partially dynamic updates (insert‑only or delete‑only).
The reduction hinges on a structural property of local marginals. After grouping edges into weight classes, the global marginal contribution of a class relative to all lower classes can be approximated by its marginal contribution within a local weight window whose aspect ratio is $O(\varepsilon^{-1})$. Summing these local marginals yields a value composition lemma that requires only approximate optimal values for each local window.
Compared with the value reduction of Gupta and Peng (FOCS 2013), whose local aspect ratio is $\varepsilon^{-\Theta(\varepsilon^{-1})}$, our approach brings the dependence down to linear $O(\varepsilon^{-1})$, a substantial improvement.
The same structural insight also leads to an improved matching composition lemma for explicit matchings. In the framework of Bernstein et al. (SODA 2025) the local aspect ratio was $O(\varepsilon^{-2})$; we reduce it to $O(\varepsilon^{-1})$, achieving better efficiency while preserving the same approximation guarantee.
Review: By carefully analyzing local marginals, this work offers a unified and tighter aspect‑ratio reduction technique. It streamlines both value‑based and matching‑based reductions, eliminates the previous exponential dependence, and lays a stronger theoretical foundation for future dynamic weighted‑matching algorithms.