Approximability of the Four-Vertex Model

TL;DR

This paper investigates the approximability of the four-vertex model, demonstrating an FPRAS under specific conditions and extending techniques to planar graphs, advancing understanding of complex vertex models.

Contribution

It introduces the first FPRAS for the four-vertex model with unwindable constraints, leveraging a worm process and linear equations over GF(2).

Findings

FPRAS exists for the four-vertex model under certain conditions

The worm process enables efficient sampling via rapid mixing

Applications extend to planar graphs for efficient sampling

Abstract

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.