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 over GF(2).The FPRAS is given by a Markov chain known as 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 unwindable constraint functions.Additionally, we explore the applications of this technique on planar graphs, providing efficient sampling algorithms.

翻译：我们研究四顶点模型（六顶点模型的特例）的可逼近性。尽管在最坏情况下该模型是NP难逼近问题，但我们证明当输入满足特定GF(2)线性方程组时，四顶点模型存在完全多项式随机近似方案（FPRAS）。该FPRAS由称为"虫形过程"的马尔可夫链实现，其状态空间与快速混合性质取决于线性方程组的解。这是首个为具有不可撤销约束函数的六顶点模型设计FPRAS的尝试。此外，我们探索了该技术在平面图中的应用，并给出高效采样算法。