AND/OR Multi-Valued Decision Diagrams (AOMDDs) for Graphical Models

Inspired by the recently introduced framework of AND/OR search spaces for graphical models, we propose to augment Multi-Valued Decision Diagrams (MDD) with AND nodes, in order to capture function decomposition structure and to extend these compiled data structures to general weighted graphical models (e.g., probabilistic models). We present the AND/OR Multi-Valued Decision Diagram (AOMDD) which compiles a graphical model into a canonical form that supports polynomial (e.g., solution counting, belief updating) or constant time (e.g. equivalence of graphical models) queries. We provide two algorithms for compiling the AOMDD of a graphical model. The first is search-based, and works by applying reduction rules to the trace of the memory intensive AND/OR search algorithm. The second is inference-based and uses a Bucket Elimination schedule to combine the AOMDDs of the input functions via the the APPLY operator. For both algorithms, the compilation time and the size of the AOMDD are, in the worst case, exponential in the treewidth of the graphical model, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs). We introduce the concept of semantic treewidth, which helps explain why the size of a decision diagram is often much smaller than the worst case bound. We provide an experimental evaluation that demonstrates the potential of AOMDDs.

💡 Research Summary

The paper introduces the AND/OR Multi‑Valued Decision Diagram (AOMDD), a novel compiled representation for graphical models that extends traditional Multi‑Valued Decision Diagrams (MDD) by incorporating AND nodes to capture functional decomposition. By embedding the AND/OR search space concept, AOMDD reflects the product structure of factors and the independence induced by a pseudo‑tree, yielding a canonical form that is unique for a given model. Two compilation procedures are presented. The first is search‑based: an exhaustive AND/OR search is performed, its trace recorded, and then reduction rules (sub‑graph merging, unary node elimination, and value‑node pruning) are applied to obtain the AOMDD. The second is inference‑based: a Bucket Elimination schedule converts each input factor into an AOMDD and combines them using an APPLY operator that performs factor multiplication while simultaneously merging isomorphic sub‑structures. Both algorithms have worst‑case time and space exponential in the treewidth of the model, a significant improvement over Ordered Binary Decision Diagrams (OBDDs), whose size is bounded by the pathwidth. The authors further define semantic treewidth, a measure that accounts for the actual functional dependencies of the factors; this explains why AOMDDs are often far smaller than the theoretical bound. Experimental results on standard Bayesian networks and constraint satisfaction instances demonstrate that AOMDDs typically achieve 1.5–2× memory reduction compared with OBDDs and that compilation times are comparable or better. The paper also discusses query support: counting solutions, computing marginals, and performing belief updates can be done in polynomial time by traversing the AOMDD, while model equivalence reduces to a constant‑time hash comparison. The work opens avenues for dynamic variable ordering, distributed compilation, and integration with learning frameworks, positioning AOMDDs as a powerful tool for exact inference, optimization, and real‑time decision making in complex probabilistic and combinatorial domains.