Incremental (1-ε)-approximate dynamic matching in O(poly(1/ε)) update time
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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 | ≥μ(G) ·α (respectively |M| ≥μ(G) ·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· 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.
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