Learning Polytrees

TL;DR

This paper investigates learning polytrees from data, showing that the optimal branching provides a good approximation but the problem is NP-hard to solve exactly or approximately.

Contribution

It proves the optimal branching is a good approximation and establishes NP-hardness of learning polytrees within any constant factor.

Findings

Optimal branching approximates the best polytree well.

Learning polytrees is NP-hard to approximate within some constant.

The problem's computational difficulty limits practical algorithms.

Abstract

We consider the task of learning the maximum-likelihood polytree from data. Our first result is a performance guarantee establishing that the optimal branching (or Chow-Liu tree), which can be computed very easily, constitutes a good approximation to the best polytree. We then show that it is not possible to do very much better, since the learning problem is NP-hard even to approximately solve within some constant factor.