  In this paper, we study the mixing time of Markov Chain Monte Carlo (MCMC)
for integer least-square (LS) optimization problems. It is found that the
mixing time of MCMC for integer LS problems depends on the structure of the
underlying lattice. More specifically, the mixing time of MCMC is closely
related to whether there is a local minimum in the lattice structure. For some
lattices, the mixing time of the Markov chain is independent of the
signal-to-noise ($SNR$) ratio and grows polynomially in the problem dimension;
while for some lattices, the mixing time grows unboundedly as $SNR$ grows. Both
theoretical and empirical results suggest that to ensure fast mixing, the
temperature for MCMC should often grow positively as the $SNR$ increases. We
also derive the probability that there exist local minima in an integer
least-square problem, which can be as high as
$1/3-\frac{1}{\sqrt{5}}+\frac{2\arctan(\sqrt{5/3})}{\sqrt{5}\pi}$.
