随机演化图中的匹配
Matching in Stochastically Evolving Graphs
摘要：
研究随机演化图的最大基数匹配问题。我们正式定义了随机出发的到达-出发模型。在那里，从特定的概率分布中采样一个图形，并将其显示为一系列快照。我们的目标是研究在采样图中创建大匹配的算法。我们定义了这个问题的随机性的价格，直观地捕捉到任何算法的损失，在最坏的情况下，由于模型的不确定性的匹配的大小。此外，我们证明了存在一个确定性的最优算法的问题。在我们的第二组结果中，我们表明，我们可以有效地近似的最大基数匹配的预期大小，通过推导出一个完全随机化的近似方案（FPRAS）为it. The FPRAS是骨干的概率算法，是最佳的模型定义在两个时间步。我们的最后一个结果是$\frac{2}{3}$的随机价格的上界。这意味着事后看来，没有算法可以匹配超过$\frac{2}{3}$的最佳匹配边缘。
英文摘要：
This paper studies the maximum cardinality matching problem in stochastically evolving graphs. We formally define the arrival-departure model with stochastic departures. There, a graph is sampled from a specific probability distribution and it is revealed as a series of snapshots. Our goal is to study algorithms that create a large matching in the sampled graphs. We define the price of stochasticity for this problem which intuitively captures the loss of any algorithm in the worst case in the size of the matching due to the uncertainty of the model. Furthermore, we prove the existence of a deterministic optimal algorithm for the problem. In our second set of results we show that we can efficiently approximate the expected size of a maximum cardinality matching by deriving a fully randomized approximation scheme (FPRAS) for it. The FPRAS is the backbone of a probabilistic algorithm that is optimal when the model is defined over two timesteps. Our last result is an upper bound of $\frac{2}{3}$ on the price of stochasticity. This means that there is no algorithm that can match more than $\frac{2}{3}$ of the edges of an optimal matching in hindsight.
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