We connect the mixing behaviour of random walks over a graph to the power of the local-consistency algorithm for the solution of the corresponding constraint satisfaction problem (CSP). We extend this connection to arbitrary CSPs and their promise variant. In this way, we establish a linear-level (and, thus, optimal) lower bound against the local-consistency algorithm applied to the class of aperiodic promise CSPs. The proof is based on a combination of the probabilistic method for random Erd\H{o}s-R\'enyi hypergraphs and a structural result on the number of fibers (i.e., long chains of hyperedges) in sparse hypergraphs of large girth. As a corollary, we completely classify the power of local consistency for the approximate graph homomorphism problem by establishing that, in the nontrivial cases, the problem has linear width.

翻译：我们将图上随机游走的混合行为与局部一致性算法求解相应约束满足问题（CSP）的能力联系起来。我们将这种联系推广到任意CSP及其承诺变体。通过这种方式，我们针对应用于非周期承诺CSP类别的局部一致性算法，建立了一个线性级别（从而是最优的）下界。该证明基于随机Erd\H{o}s-R\'enyi超图的概率方法，以及大围长稀疏超图中纤维（即超边的长链）数量的结构结果。作为推论，我们通过证明在非平凡情况下该问题具有线性宽度，完全分类了局部一致性算法在近似图同态问题上的能力。