Problem 39: Approximating Maximum Matching Size

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Suggested by Krzysztof Onak
Source Bertinoro 2011
Short link https://sublinear.info/39

Consider graphs with maximum degree bounded by $d$. It is possible to approximate the size of the maximum matching up to an additive $\epsilon n$ in time that is a function of only $\epsilon$ and $d$ [NguyenO-08,YoshidaYI-09]. The fastest currently known algorithm runs in $d^{O(1/\epsilon^2)}$ time [YoshidaYI-09].

Question: Is there an algorithm that runs in $\operatorname{poly}(d/\epsilon)$ time?