Incremental(1-∊) -approximate dynamic matching inO(poly(1/∊))update time - Blikstad, Joakim; Kiss, Peter (2023)
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.