随机图的谱种植与反驳割、可着色性和社区的困难性
Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
摘要：
研究了有效地反驳图的k-可染性，或等价地证明其色数的下界的问题。我们给这个问题的平均情况下的计算硬度在稀疏随机正则图的正式证据，显示一个简单的频谱证书的最优性。这一证据采取的形式计算安静种植：我们构建了一个分布的d-正则图，具有显着较小的色数比一个典型的定期均匀随机绘制的图，同时提供证据表明，这两个分布是无法区分的一大类算法。我们将我们的结果推广到更一般的问题，证明最大k-截上界。这种安静的种植是通过最小化种植结构（例如着色或切割）对图谱的影响来实现的。具体地，种植结构精确地对应于邻接矩阵的特征向量。这避免了随机矩阵理论的推出效应，并且延迟了种植在光谱或局部统计中变得可见的点。为了进一步说明这一点，我们给出了类似的结果，高斯模拟这个问题：一个安静的版本的尖峰模型，在那里我们植物的特征空间，而不是添加一个通用的低秩扰动。我们的证据区分两种分布的计算难度是基于三个不同的算法：稳定的信念传播，本地统计层次结构，和低度似然比。独立的兴趣，我们的研究结果包括多尖峰矩阵模型的低度似然比的通用界，和改进的随机块模型的低度分析。
英文摘要：
We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model.
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