On the Parameterized Complexity of Polytree Learning

TL;DR

This paper analyzes the computational complexity of learning polytrees in Bayesian networks, providing new algorithms and complexity bounds depending on the number of variables and parent set sizes.

Contribution

It establishes the fixed-parameter tractability of Polytree Learning with respect to total variables and parent set size, and shows hardness results related to the number of variables receiving parents.

Findings

Polytree Learning can be solved in $3^n imes |I|^{O(1)}$ time.

No fixed-parameter tractable algorithm exists when parameterized by the number of variables with nonempty parent sets.

Efficient algorithms are possible when both the number of such variables and maximum parent set size are bounded.

Abstract

A Bayesian network is a directed acyclic graph that represents statistical dependencies between variables of a joint probability distribution. A fundamental task in data science is to learn a Bayesian network from observed data. \textsc{Polytree Learning} is the problem of learning an optimal Bayesian network that fulfills the additional property that its underlying undirected graph is a forest. In this work, we revisit the complexity of \textsc{Polytree Learning}. We show that \textsc{Polytree Learning} can be solved in $3^n \cdot |I|^{\mathcal{O}(1)}$ time where $n$ is the number of variables and $|I|$ is the total instance size. Moreover, we consider the influence of the number of variables $d$ that might receive a nonempty parent set in the final DAG on the complexity of \textsc{Polytree Learning}. We show that \textsc{Polytree Learning} has no $f(d)\cdot |I|^{\mathcal{O}(1)}$-time algorithm, unlike Bayesian network learning which can be solved in $2^d \cdot |I|^{\mathcal{O}(1)}$ time. We show that, in contrast, if $d$ and the maximum parent set size are bounded, then we can obtain efficient algorithms.