Order Statistics for Probabilistic Graphical Models

We consider the problem of computing r-th order statistics, namely finding an assignment having rank r in a probabilistic graphical model. We show that this problem is NP-hard even when the graphical model has no edges (zero-treewidth models) via a reduction from the number partitioning problem. We use this reduction, specifically pseudopolynomial time algorithms for number partitioning, to yield a pseudo-polynomial time approximation algorithm for solving the r-th order statistics problem in zero-treewidth models. We then extend this algorithm to general graphical models by generalizing it to tree decompositions, and demonstrate via experimental evaluation on various datasets that our proposed algorithm is more accurate than sampling algorithms for computing r-th order statistics.