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arXiv 1206.2269cs.DS
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流模型中匹配的更好边界

Better bounds for matchings in the streaming model

Michael Kapralov

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摘要：

在本文中，我们提出了改进的边界近似最大匹配的二部图中的流模型。首先，我们考虑的问题，以及最大匹配可以近似在一个单一的通过输入使用$\tilde O(n)$空间，其中$n$是在输入图中的顶点数。在文献中已经考虑了这个问题的两个自然变体：（1）边到达设置，其中边到达流中，以及（2）顶点到达设置，其中图的一侧上的顶点与它们的所有入射边一起到达流中。后一种设置也在在线算法的上下文中得到了广泛的研究，其中每个到达的顶点必须在到达时被匹配或丢弃。在在线设置中，著名的Karp-Vazirani-Vazirani算法实现了$1-1/e$近似。尽管流模型的限制较少，因为算法不被约束为在到达时可匹配地匹配顶点，但是在具有顶点到达和$\tilde O(n)$空间的流模型中最公知的近似是$1-1/e$的相同因子。我们表明，没有一个单通流算法，使用$\tilde O(n)$空间可以实现一个更好的比$1-1/e$近似最大匹配，即使在顶点到达设置。这导致了一个惊人的结论，即没有一个单通道流算法可以做得比在线算法更好，除非它使用明显超过$\tilde O(n)$空间。此外，我们的边界产生最知名的不可能结果近似匹配的边缘到达模型。我们还给出了一个简单的算法，实现近似比$1-e^{-k}k^{k-1}/(k-1)!=1-\frac1{\sqrt{2πk}}+o(1/k)$在$k$通过顶点到达模型使用线性空间，改进以前最好的已知收敛。

英文摘要：

In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input using $\tilde O(n)$ space, where $n$ is the number of vertices in the input graph. Two natural variants of this problem have been considered in the literature: (1) the edge arrival setting, where edges arrive in the stream and (2) the vertex arrival setting, where vertices on one side of the graph arrive in the stream together with all their incident edges. The latter setting has also been studied extensively in the context of online algorithms, where each arriving vertex has to either be matched irrevocably or discarded upon arrival. In the online setting, the celebrated algorithm of Karp-Vazirani-Vazirani achieves a $1-1/e$ approximation. Despite the fact that the streaming model is less restrictive in that the algorithm is not constrained to match vertices irrevocably upon arrival, the best known approximation in the streaming model with vertex arrivals and $\tilde O(n)$ space is the same factor of $1-1/e$. We show that no single pass streaming algorithm that uses $\tilde O(n)$ space can achieve a better than $1-1/e$ approximation to maximum matching, even in the vertex arrival setting. This leads to the striking conclusion that no single pass streaming algorithm can do better than online algorithms unless it uses significantly more than $\tilde O(n)$ space. Additionally, our bound yields the best known impossibility result for approximating matchings in the edge arrival model. We also give a simple algorithm that achieves approximation ratio $1-e^{-k}k^{k-1}/(k-1)!=1-\frac1{\sqrt{2πk}}+o(1/k)$ in $k$ passes in the vertex arrival model using linear space, improving upon previously best known convergence.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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