Learning Chordal Markov Networks via Branch and Bound
We present a new algorithmic approach for the task of finding a chordal Markov network structure that maximizes a given scoring function. The algorithm is based on branch and bound and integrates dynamic programming for both domain pruning and for obtaining strong bounds for search-space pruning. Empirically, we show that the approach dominates in terms of running times a recent integer programming approach (and thereby also a recent constraint optimization approach) for the problem. Furthermore, our algorithm scales at times further with respect to the number of variables than a state-of-the-art dynamic programming algorithm for the problem, with the potential of reaching 20 variables and at the same time circumventing the tight exponential lower bounds on memory consumption of the pure dynamic programming approach.
The authors present a branch and bound algorithm for learning Chordal Markov networks. The prior state of the art algorithm is a dynamic programming approach based on a recursive characterization of clique tress and storing in memory the scores of already-solved subproblems. The proposed algorithm uses a branch and bound algorithm to search for an optimal chordal Markov network. The algorithm first uses a dynamic programming algorithm to enumerate Bayesian network structures, which are later used as pruning bounds. A symmetry breaking technique is introduced to prune the search space. 
The special property of Chordal Markov networks necessitates that no score pruning can be done. Given that this step is exponential in time and space. It is unlikely that learning CMSL can scale to even medium-size datasets (>20-30variables). 
What is the definition of decomposable DAG? This is important for understanding other concepts defined later on, such as ordered decomposable DAG.
The symmetry breaking rule is useful. What is its relation to the symmetry breaking rules introduced in (van Beek&Hoffmann 15) for learning BNs? The authors referenced that paper and should have discussed the relevance.
The paper did not give a clear picture on how much time is spent on computing scores, and how much time is spent on optimization. Also, the total runtime of the proposed algorithm includes computing optimal BNs and segment trees. How much do they take relative to the total runtime?  The aggregated time does not provide a complete picture.