An estimator for matching size in low arboricity graphs with two applications

Abstract

In this paper, we present a new 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 \(O\left( \frac{\sqrt{n}}{\varepsilon ^2}\log n\right) \) space algorithm for approximating the matching size of a planar graph on n vertices within \((3.5+\varepsilon )\) factor. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within \((3.5+\varepsilon )\) factor using \(O\left( \frac{n^{2/3}}{\varepsilon ^2}\log n\right) \) communication from each player. In comparison with previous works, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor in the case of planar graphs.