Structure Learning of H-Colorings

Abstract

We study the following structure learning problem for H -colorings. For a fixed (and known) constraint graph H with q colors, given access to uniformly random H -colorings of an unknown graph G=(V,E) , how many samples are required to learn the edges of  G ? We give a characterization of the constraint graphs  H for which the problem is identifiable for every  G and show that there are identifiable constraint graphs for which one cannot hope to learn every graph  G efficiently. We provide refined results for the case of proper vertex q -colorings of graphs of maximum degree  d . In particular, we prove that in the tree uniqueness region (i.e., when q≤ d ), the problem is identifiable and we can learn G in poly( d,q )× O(n 2 log n ) time. In the tree non-uniqueness region (i.e., when q≤ d), we show that the problem is not identifiable and thus G cannot be learned. Moreover, when q ≤ d - √d + Θ (1), we establish that even learning an equivalent graph (any graph with the same set of H -colorings) is computationally hard—sample complexity is exponential in n in the worst case. We further explore the connection between the efficiency/hardness of the structure learning problem and the uniqueness/non-uniqueness phase transition for general H -colorings and prove that under a well-known uniqueness condition in statistical physics, we can learn G in poly( d,q )× O(n 2 log n ) time.