Approximability of the Four-Vertex Model

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.