Parameterized Complexity Results for Exact Bayesian Network Structure Learning

Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) super-structure. The super-structure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed super-structure as a natural generalization of its undirected counterpart. Our results apply to several variants of score-based Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth, and in linear time if in addition the super-structure has bounded maximum degree. Furthermore, we show that if the directed super-structure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Similarly, exact Bayesian network structure learning remains NP-hard for “almost acyclic” directed super-structures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).

💡 Research Summary

Bayesian network structure learning (BNSL) is a central problem in probabilistic graphical models: given a data set, one must select a directed acyclic graph (DAG) that maximises a chosen scoring function. The exact version—searching for the globally optimal DAG—is well‑known to be NP‑hard. This paper adopts a parameterised‑complexity viewpoint and asks how structural restrictions on a so‑called “super‑structure” affect the worst‑case computational cost of exact BNSL.

The super‑structure is an undirected graph that contains the skeletons (the underlying undirected edges) of all feasible solution networks. The authors introduce a natural generalisation, the directed super‑structure, which records for each potential arc whether it is allowed in a particular direction. The analysis assumes a decomposable score: the total score of a network is the sum of local scores, each depending only on a node and its parent set. This decomposition is the key to applying dynamic programming (DP) on tree decompositions.

Positive algorithmic results

1. Bounded treewidth – If the (undirected) super‑structure has treewidth ≤ k for some constant k, the authors show that exact BNSL can be solved in non‑uniform polynomial time. Using a tree decomposition of width k, a DP processes each bag, enumerating all admissible parent sets for the vertices in the bag and propagating the best partial scores upward. The running time is O(n·f(k)), where f(k) is exponential in k but independent of n, yielding a fixed‑parameter tractable (FPT) algorithm when k is treated as a parameter.
2. Bounded degree in addition – When the super‑structure also has maximum degree Δ bounded by a constant, the DP’s search space shrinks dramatically because each node can have only O(Δ) possible parents. By pre‑computing tables for each bag and using adjacency‑list look‑ups, the overall algorithm runs in linear time O(m), where m is the number of edges in the super‑structure. This is a striking improvement: the combination of bounded treewidth and bounded degree yields a truly scalable exact solver.
3. Acyclic directed super‑structure – If the directed super‑structure itself is a DAG (i.e., it contains no directed cycles), the problem can be solved in quadratic time O(n²). A topological order of the vertices is obtained in O(n+m) time; then each vertex is processed in that order, evaluating at most n candidate parent sets. The absence of cycles guarantees that the DP never needs to revisit a node, leading to the O(n²) bound.

Hardness and necessity of the restrictions
The authors complement the algorithmic upper bounds with a suite of lower‑bound results that demonstrate the essential nature of the three parameters.

• Treewidth alone is insufficient for uniform polynomial time – Dropping the bounded‑degree condition, even with constant treewidth, prevents the existence of a uniform polynomial‑time algorithm unless widely believed complexity collapses occur (e.g., FPT = W