We study the approximability of the four-vertex model, a special case of the six-vertex model.We prove that, despite being NP-hard to approximate in the worst case, the four-vertex model admits a fully polynomial randomized approximation scheme (FPRAS) when the input satisfies certain linear equation system.The FPRAS is given by a Markov chain called the worm process whose state space and rapid mixing rely on the solution of the linear equation system.This is the first attempt to design an FPRAS for the six-vertex model with unwinable constraint functions.Furthermore, we consider the application of this technique on planar graphs to give efficient sampling algorithms.

翻译：我们研究了四顶点模型（六顶点模型的一个特例）的近似性。我们证明，尽管在最坏情况下该模型是NP难近似的，但当输入满足特定线性方程组时，四顶点模型承认一个完全多项式随机近似方案（FPRAS）。该FPRAS由一种称为“蠕虫过程”（worm process）的马尔可夫链给出，其状态空间与快速混合性依赖于该线性方程组的解。这是首次尝试为具有不可胜约束函数的六顶点模型设计FPRAS。此外，我们考虑将这一方法应用于平面图，以给出高效采样算法。