In this paper, we present a new simple degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by $\alpha$, the estimator gives a $\alpha+2$ factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a $3.5$ approximation of the matching size. Using this estimator, we get new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized $O(\frac{\sqrt{n}}{\epsilon^2}\log n)$ space algorithm for approximating the matching size within $(3.5+\epsilon)$ factor in a planar graph on $n$ vertices. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within $(3.5+\epsilon)$ factor using $O(\frac{n^{2/3}}{\epsilon^2}\log n)$ communication from each player. In comparison with the previous estimators, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor for the case of planar graphs.
翻译：在本文中, 我们展示了一个新的简单的基于度的测算器, 用于在受约束的偏差图中匹配最大匹配的大小。 当图形的偏差范围被 $\ ALpha$ 约束时, 估计器给出了一个匹配大小的 $\ ALpha+ 2$ 系数近似值。 对于平面图, 我们显示估计器效果更好, 并返回匹配大小的 3.5$ 近似值。 使用此估计器, 我们获得新的结果, 以接近在流和分布的计算模型中, 平面图图图的匹配大小。 特别是, 当图形的偏差范围被 $\ ALphapha$ 约束时, 我们得到一个随机化的 $ (\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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