  Inference is a main task in structured prediction and it is naturally modeled
with a graph. In the context of Markov random fields, noisy observations
corresponding to nodes and edges are usually involved, and the goal of exact
inference is to recover the unknown true label for each node precisely. The
focus of this paper is on the fundamental limits of exact recovery irrespective
of computational efficiency, assuming the generative process proposed by
Globerson et al. (2015). We derive the necessary condition for any algorithm
and the sufficient condition for maximum likelihood estimation to achieve exact
recovery with high probability, and reveal that the sufficient and necessary
conditions are tight up to a logarithmic factor for a wide range of graphs.
Finally, we show that there exists a gap between the fundamental limits and the
performance of the computationally tractable method of Bello and Honorio
(2019), which implies the need for further development of algorithms for exact
inference.
