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

中文翻译：

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超越可绕性：四顶点模型的近似性

我们研究四顶点模型的逼近性，这是六顶点模型的一个特例。我们证明，尽管在最坏情况下是 NP 难逼近的，但当输入满足某个线性方程组时，四顶点模型承认完全多项式随机逼近方案（FPRAS）。 FPRAS 由称为 的马尔可夫链给出，其状态空间和快速混合依赖于线性方程组的解。这是针对具有约束函数的六顶点模型设计FPRAS的首次尝试。此外，我们探索了该技术在平面图上的应用，提供了有效的采样算法。