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.

中文翻译：

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光谱种植和驳斥随机图中切割、可着色性和社区的硬度

我们研究了有效驳斥图的 k 可着色性的问题，或者等效地证明其色数的下界。我们在稀疏随机正则图中给出了这个问题的平均情况计算难度的正式证据，显示了简单谱证明的最优性。该证据采用计算安静种植的形式：我们构建了一个 d 正则图的分布，该分布的色数明显小于随机均匀绘制的典型正则图，同时提供证据表明这两种分布无法被大类区分的算法。我们将我们的结果推广到更一般的问题，即证明最大 k-cut 的上限。这种安静的种植是通过最大限度地减少种植结构的影响来实现的（例如 图谱上的着色或切割）。具体来说，种植结构与邻接矩阵的特征向量完全对应。这避免了随机矩阵理论的推出效应，并延迟了种植在频谱或局部统计中变得可见的点。为了进一步说明这一点，我们对这个问题的高斯类比给出了类似的结果：尖峰模型的安静版本，我们在其中植入了一个特征空间，而不是添加一个通用的低秩扰动。我们区分两个分布的计算难度的证据基于三种不同的启发式：信念传播的稳定性、局部统计层次结构和低度似然比。独立感兴趣的是，我们的结果包括多尖峰矩阵模型的低度似然比的通用界限，