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 $O(\frac{\sqrt{n}}{\eps^2}\log n)$ space algorithm for approximating the matching size within $(3.5+\eps)$ 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 $(3.5+\eps)$ factor using $O(\frac{n^{2/3}}{\eps^2}\log n)$ 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.

中文翻译：

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低树形图中具有两个应用程序的大小匹配估计器

在本文中，我们提出了一个新的基于度的简单估计器，用于有界树状图中最大匹配的大小。当图的树状度受$ \ alpha $限制时，估算器会给出匹配大小的$ \ alpha + 2 $因子近似值。对于平面图，我们显示估计器的效果更好，并且返回匹配大小的近似$ 3.5 $。使用该估计器，我们得到了新的结果，用于近似计算流式和分布式计算模型中平面图的匹配大小。特别是在到达顶点的流中，我们获得了一个随机的$ O（\ frac {\ sqrt {n}} {\ eps ^ 2} \ log n）$空间算法，用于近似匹配$（3.5+ \ eps）$在$ n $个顶点上的平面图中的因子。类似地，我们在顶点分区模型中获得了一个同时协议，用于近似$（3。使用来自每个玩家的$ O（\ frac {n ^ {2/3}} {\ eps ^ 2} \ log n）$沟通获得5+ \ eps）$的系数。与先前的估计器相比，本文中的估计器不需要知道输入图的拟真度，并提高了平面图情况下的近似因子。