Some difficult computation problems, depicted by finding the highest peak in a "landscape" of countless mountain peaks separated by valleys, can take advantage of the Overlap Gap Property: At a high enough "altitude," any two points will be either close or far apart -- but nothing in-between. David Gamarnik has developed a new tool, the Overlap Gap Property, for understanding computational problems that appear intractable. The notion that some computational problems in math and computer science can be hard should come as no surprise. There is, in fact, an entire class of problems deemed impossible to solve algorithmically. Just below this class lie slightly "easier" problems that are less well-understood -- and may be impossible, too.

In this paper we consider the problem of learning undirected graphical models from data generated according to the Glauber dynamics. The Glauber dynamics is a Markov chain that sequentially updates individual nodes (variables) in a graphical model and it is frequently used to sample from the stationary distribution (to which it converges given sufficient time). Additionally, the Glauber dynamics is a natural dynamical model in a variety of settings. This work deviates from the standard formulation of graphical model learning in the literature, where one assumes access to i.i.d. samples from the distribution. Much of the research on graphical model learning has been directed towards finding algorithms with low computational cost. As the main result of this work, we establish that the problem of reconstructing binary pairwise graphical models is computationally tractable when we observe the Glauber dynamics. Specifically, we show that a binary pairwise graphical model on $p$ nodes with maximum degree $d$ can be learned in time $f(d)p^2\log p$, for a function $f(d)$, using nearly the information-theoretic minimum number of samples.