In the dynamic approximate maximum bipartite matching problem we are given bipartite graph $G$ undergoing updates and our goal is to maintain a matching of $G$ which is large compared the maximum matching size $\mu(G)$. We define a dynamic matching algorithm to be $\alpha$ (respectively $(\alpha, \beta)$)-approximate if it maintains matching $M$ such that at all times $|M | \geq \mu(G) \cdot \alpha$ (respectively $|M| \geq \mu(G) \cdot \alpha - \beta$). We present the first deterministic $(1-\epsilon )$-approximate dynamic matching algorithm with $O(poly(\epsilon ^{-1}))$ amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or exponential in $1/\epsilon $ [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive $(1, \epsilon \cdot n)$-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for $(1-\epsilon )$-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of $G$ in a fully dynamic manner.

翻译：在动态近似最大二分匹配问题中，给定一个经历更新的二分图$G$，目标是在线维护一个与最大匹配规模$\mu(G)$相比较大的匹配$M$。我们定义动态匹配算法为$\alpha$近似（或$(\alpha, \beta)$近似），若其维护的匹配$M$始终满足$|M| \geq \mu(G) \cdot \alpha$（或$|M| \geq \mu(G) \cdot \alpha - \beta$）。针对发生边插入操作的图，我们提出了首个确定性$(1-\epsilon )$近似动态匹配算法，其摊还更新时间复杂度为$O(poly(\epsilon ^{-1}))$。此前解决方案需要超常数时间[Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23]或$1/\epsilon$指数级时间[Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19]。我们的实现比上述算法更为简洁，且描述是自包含的。进一步地，我们证明在允许加法$(1, \epsilon \cdot n)$近似的条件下，该算法可无缝扩展以处理边插入之外的顶点删除操作。这使得我们的算法成为少数能以$O(小数值)$更新时间处理$(1-\epsilon)$近似动态匹配的算法之一，能够以完全动态方式应对同时增大和减小$G$最大匹配规模的更新。