Learning Minimum-Size BDDs: Towards Efficient Exact Algorithms

Binary decision diagrams (BDDs) are widely applied tools to compactly represent labeled data as directed acyclic graphs; for efficiency and interpretability reasons small BDDs are preferred.Given labeled data, minimizing BDDs is NP-complete and thus recent research focused on the influence of parameters such as the solution size $s$ on the complexity [Ordyniak et al., AAAI 2024].Our main positive result is an algorithm that is efficient if in particular $s$, the domain size $D$, and the Hamming distance between any two data points is small, improving on previous running-time bounds.This algorithm is inspired by the witness-tree paradigm that was recently successful for computing decision trees [Komusiewicz et al., ICML 2023], whose extension to BDDs was open.We extend our algorithmic results to the case where we allow a small number of misclassified data points and complement them with lower bounds that show that the running times are tight from multiple points of view.We show that our main algorithm holds practical promise by providing a proof-of-concept implementation.

This paper introduces a new algorithm, called WitBDD, for efficiently building small binary decision diagrams (BDDs), which are models used in data classification and AI. BDDs, a generalization of decision trees, are popular due to their simplicity and understandability. Keeping BDDs small is important to ensure efficiency and for allowing the model to be interpreted by humans. The problem of finding a smallest binary decision diagram is, however, very hard (NP-complete).We adapt a method known as the "witness paradigm"—previously successful with decision trees—to BDDs. This approach gradually refines a BDD by identifying misclassified data points and adding necessary corrections, greatly reducing the number of possibilities the algorithm has to explore. We prove mathematically that the algorithm is fast when the desired BDD size and data differences are small and also extend it to allow for classification errors. In experiments, our new algorithm outperforms existing SAT-based methods for small BDDs (sizes $\leq 4$), solving problems faster in many cases. For larger BDDs, existing methods are still better.In short, our work makes computing small, optimal BDDs more practical and opens the door for further optimization and use in explainable AI.