Abstract
We present two fully dynamic algorithms for maximum cardinality matching in bipartite graphs. Our main result is a deterministic algorithm that maintains a (3/2+ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(3/2 + \epsilon )$$\end{document} approximation in worst-case update time O(m1/4ϵ-2.5)\documentclass[12pt]{minimal}
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\begin{document}$$O(m^{1/4}\epsilon ^{-2.5})$$\end{document}. This algorithm is polynomially faster than all previous deterministic algorithms for any constant approximation, and faster than all previous algorithms (randomized included) that achieve a better-than-2 approximation. We also give stronger results for bipartite graphs whose arboricity is at most α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, achieving a (1+ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(1+ \epsilon )$$\end{document} approximation in worst-case update time O(α(α+log(n))+ϵ-4(α+log(n))+ϵ-6)\documentclass[12pt]{minimal}
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\begin{document}$$O(\alpha (\alpha + \log (n)) + \epsilon ^{-4}(\alpha + \log (n)) + \epsilon ^{-6})$$\end{document}, which is O(α(α+logn))\documentclass[12pt]{minimal}
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\begin{document}$$O(\alpha (\alpha + \log n))$$\end{document} for constant ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon $$\end{document}. Previous results for small arboricity graphs had similar update times but could only maintain a maximal matching (2-approximation). All these previous algorithms, however, were not limited to bipartite graphs.