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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper presents two very natural methods for searching for Bayesian networks of treewidth <= k. The exact method uses a MILP to search jointly over (Bayesian network, elimination order, chordal graph induced by elimination on the moralized network). The approximate method repeatedly samples a random k-tree and then uses dynamic programming or sampling to find the best network whose moralization is a subgraph of the k-tree (any network of treewidth <= k can be obtained in this way). The paper is clear and the methods seem like textbook material.