  The six-vertex model is an important model in statistical physics and has
deep connections with counting problems. There have been some fully polynomial
randomized approximation schemes (FPRAS) for the six-vertex model [30, 10],
which all require that the constraint functions are windable. In the present
paper, we give an FPRAS for the six-vertex model with an unwindable constraint
function by Markov Chain Monte Carlo method (MCMC). Different from [10], we use
the Glauber dynamics to design the Markov Chain depending on a circuit
decomposition of the underlying graph. Moreover, we prove the rapid mixing of
the Markov Chain by coupling, instead of canonical paths in [10].
