Incremental $(1-ε)$-approximate dynamic matching in $O(poly(1/ε))$ update time

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 $μ(G)$. We define a dynamic matching algorithm to be $α$ (respectively $(α, β)$)-approximate if it maintains matching $M$ such that at all times $|M | \geq μ(G) \cdot α$ (respectively $|M| \geq μ(G) \cdot α- β$). We present the first deterministic $(1-ε)$-approximate dynamic matching algorithm with $O(poly(ε^{-1}))$ 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 $1/ε$ [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 $(1, ε\cdot n)$-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 $(1-ε)$-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of $G$ in a fully dynamic manner.