The Complexity of Bayesian Network Learning: Revisiting the Superstructure
This study investigates the parameterized complexity of Bayesian network structure learning (BNSL) under super-structure constraints. By integrating graph-theoretic parameters—such as feedback edge set size, local feedback edge set size, and treewidth—with input representation formats, particularly additive representations, the work systematically analyzes the fixed-parameter tractability of BNSL. The main contributions include the first proof that BNSL is fixed-parameter tractable when parameterized by the size of a feedback edge set, a result extended to local feedback edge sets. Furthermore, it establishes that under additive representations, treewidth alone suffices for fixed-parameter tractability—a finding that also applies to Polytree learning. The paper provides a complete complexity classification across mainstream graph parameters and derives corresponding conditional lower bounds, thereby significantly advancing the theoretical foundation of BNSL.