论文标题
通过查询访问最大匹配的Oracle构建大型匹配
Constructing Large Matchings via Query Access to a Maximal Matching Oracle
论文作者
论文摘要
Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in两部分图形和$ o（（（\ frac {1}ε）^{\ frac {\ frac {1}ε}）$或$ o（\ text {poly}（\ frac {1}ε）\ cdot \ cdot \ log n）$ pass in Penteral Graphs中，其中$ n $ n $ n $ n $ n $ n属于Vertipe of Vertipe of Vertipe of Vertipe of Vertipt。但是，到目前为止，证明这种算法的不可能结果是难以捉摸的，例如，即使存在$ 0.999 $ $ 0.999 $的$ 2 $ pass小空间流算法。 最大匹配的所有多通流算法的关键构建块是贪婪的匹配算法。我们的目的是了解这种方法的局限性：如果算法仅依赖于贪婪算法的调用，则需要多少次通过？ 在本文中，我们启动了多通河流算法限制家族的下限研究，以实现最大匹配。我们专注于简单但功能强大的算法类别，这些算法中的每个通行证在输入图的顶点诱导的子图上贪婪。在双方图中，我们表明$ 3 $通过是必要的，足以提高$ 1/2 $的微不足道近似系数：我们给出了此类算法的近似值的下限$ 0.6 $，这是最佳的。我们进一步表明，即使在双方图中，计算$（1-ε）$近似值也需要$ω（\ frac {1}ε）$ passes。最后，所考虑的算法类别不适合一般图表：我们表明，为了提高$ 1/2 $的微不足道近似系数，需要$ω（n）$ passes。
Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in bipartite graphs and in $O( (\frac{1}ε)^{\frac{1}ε})$ or $O(\text{poly} (\frac{1}ε) \cdot \log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for example, even the existence of $2$-pass small space streaming algorithms with approximation factor $0.999$ has not yet been ruled out. The key building block of all multi-pass streaming algorithms for Maximum Matching is the Greedy matching algorithm. Our aim is to understand the limitations of this approach: How many passes are required if the algorithm solely relies on the invocation of the Greedy algorithm? In this paper, we initiate the study of lower bounds for restricted families of multi-pass streaming algorithms for Maximum Matching. We focus on the simple yet powerful class of algorithms that in each pass run Greedy on a vertex-induced subgraph of the input graph. In bipartite graphs, we show that $3$ passes are necessary and sufficient to improve on the trivial approximation factor of $1/2$: We give a lower bound of $0.6$ on the approximation ratio of such algorithms, which is optimal. We further show that $Ω( \frac{1}ε)$ passes are required for computing a $(1-ε)$-approximation, even in bipartite graphs. Last, the considered class of algorithms is not well-suited to general graphs: We show that $Ω(n)$ passes are required in order to improve on the trivial approximation factor of $1/2$.
