Exact and Approximate Algorithms for Polytree Learning

2026-05-05

AI Summary

This work addresses the NP-hard problem of learning polytree structures, systematically investigating the computational limits while preserving inference efficiency and interpretability. By incorporating indegree constraints, analyzing score function properties, and leveraging parameterized algorithms, combinatorial optimization, and approximation theory, the study presents an exact algorithm with time complexity $O((2+\varepsilon)^n)$, substantially improving upon the previous best-known bound of $O(3^n)$. Moreover, it introduces the first polynomial-time approximation algorithms: a $k$-approximation for general scoring functions and a tight 2-approximation for additive scores. Theoretical analysis demonstrates that most of these results are nearly optimal, establishing fundamental performance guarantees for polytree structure learning under practical constraints.

Abstract

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the problem of learning the best polytree is NP-hard, we study which restrictions make it more tractable by considering for example in-degree bounds, properties of score functions measuring the quality of a polytree, and approximation algorithms. We devise an algorithm that finds the optimal polytree in time $O((2+ε)^n)$ for arbitrarily small $ε> 0$ and any constant in-degree bound $k$, improving over the fastest previously known algorithm of time complexity $O(3^n)$. We further give polynomial-time algorithms for finding a polytree whose score is within a factor of $k$ from the optimal one for arbitrary scores and a factor of $2$ for additive ones. Many of the results are complemented by (nearly) tight lower bounds for either the time complexity or the approximation factors.

Problem

• polytree learning
• Bayesian networks
• NP-hard
• score functions
• in-degree bounds

Innovation

• polytree learning
• exact algorithm
• approximation algorithm
• NP-hard
• Bayesian networks