Exact and Approximate Algorithms for Polytree Learning

• URL: http://arxiv.org/abs/2605.03622v1
• Date: Tue, 05 May 2026 10:50:14 GMT
• Title: Exact and Approximate Algorithms for Polytree Learning
• Abstract summary: Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest.<n>Since the problem of learning the best polytree is NP-hard, we study which restrictions make it more tractable.<n>We devise an algorithm that finds the optimal polytree in time $O((2+)n)$ for arbitrarily small $> 0$ and any constant in-degree bound.
• Score: 9.426097667758627
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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