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图匹配问题中，给定一个经历更新的二分图$G$，我们的目标是维护一个与最大匹配规模$\mu(G)$相比足够大的匹配$M$。我们定义动态匹配算法为$\alpha$（相应地$(\alpha, \beta)$）-近似，如果它维护的匹配$M$始终满足$|M| \geq \mu(G) \cdot \alpha$（相应地$|M| \geq \mu(G) \cdot \alpha - \beta$）。针对边插入操作的图，我们提出了首个具有$O(poly(\epsilon ^{-1}))$均摊更新时间的确定性$(1-\epsilon )$-近似动态匹配算法。先前的工作要么需要超常数更新时间[Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23]，要么需要关于$1/\epsilon$的指数级更新时间[Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19]。我们的实现比上述算法更简洁，且其描述具有自包含性。此外，我们证明若允许加性$(1, \epsilon \cdot n)$-近似，我们的算法可无缝扩展至同时处理边插入和顶点删除操作。这使得本算法成为少数具有小更新时间的$(1-\epsilon )$-近似动态匹配算法之一，能够以完全动态方式处理同时增加和减小$G$最大匹配规模的更新操作。