谱图二分的新能力和局限性
New Abilities and Limitations of Spectral Graph Bisection
摘要：
基于谱的算法是一种常用的算法，当最优性不能被证明时，该算法可以确定可证明的最小图二分或输出“失败”。在本文中，我们专注于Boppana的算法，属于这种类型的最突出的方法之一。该算法在随机图类\emph{planted bisection model} --分析最小二分法及相关问题的标准图类中有很好的应用。在2001年Feige和Kilian提出的问题，如果Boppana的算法在Blum和Spencer的半随机模型中工作良好。在我们的论文中，我们肯定地回答了这个问题。我们还表明，该算法实现了类似的性能图类扩展的半随机模型。由于Boppana算法在半随机图上的行为仍然是未知的，Feige和Kilian提出了一种新的基于半定规划（SDP）的方法，并证明了它在这个模型上是有效的。基于SDP的算法和Boppana的方法的性能之间的关系被留下作为一个开放的问题。本文通过证明Feige和Kilian的二分算法与Boppana的算法提供了完全相同的结果，完整地解决了这个问题。因此，我们得到，Boppana的算法达到了最佳阈值的精确集群恢复在\emph{stochastic block model}。另一方面，我们证明了一些局限性Boppana的方法：我们表明，如果密度差的种植二分法模型的参数是太小，那么该算法失败的概率很高的模型。
英文摘要：
Spectral based heuristics belong to well-known commonly used methods which determines provably minimal graph bisection or outputs "fail" when the optimality cannot be certified. In this paper we focus on Boppana's algorithm which belongs to one of the most prominent methods of this type. It is well known that the algorithm works well in the random \emph{planted bisection model} -- the standard class of graphs for analysis minimum bisection and relevant problems. In 2001 Feige and Kilian posed the question if Boppana's algorithm works well in the semirandom model by Blum and Spencer. In our paper we answer this question affirmatively. We show also that the algorithm achieves similar performance on graph classes which extend the semirandom model. Since the behavior of Boppana's algorithm on the semirandom graphs remained unknown, Feige and Kilian proposed a new semidefinite programming (SDP) based approach and proved that it works on this model. The relationship between the performance of the SDP based algorithm and Boppana's approach was left as an open problem. In this paper we solve the problem in a complete way by proving that the bisection algorithm of Feige and Kilian provides exactly the same results as Boppana's algorithm. As a consequence we get that Boppana's algorithm achieves the optimal threshold for exact cluster recovery in the \emph{stochastic block model}. On the other hand we prove some limitations of Boppana's approach: we show that if the density difference on the parameters of the planted bisection model is too small then the algorithm fails with high probability in the model.
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