An Approximate Formulation based on Integer Linear Programming for Learning Maximum Weighted (k+1)-order Decomposable Graphs

In this work we deal with the problem of learning a maximum weighted (k + 1)order decomposable graph coarser than a given maximal k-order decomposable graph. An Integer Linear Programming formulation for the problem has been recently proposed and used in order to solve instances of the problem with a moderate number of vertices [8]. However, as the problem is known to be NP-hard, it is of practical interest to develop approximate algorithms able to work with a limited amount of computational resources. In this paper we propose an approximate Integer Linear Programming formulation for the problem using a distance-based criteria that selects a subset of edges. This criteria discards the set of edges that, on average, have lower probability to be part of the optimal solution than any of the selected edges. Experiments have been carried out with probabilistic graphical models. Using the approximate formulation we have obtained results close to the optimum, even with small threshold distance values. The obtained good results indicate that the approximate formulation has important applications for learning probabilistic graphical models using decomposable scores.