Finding a Path is Harder than Finding a Tree

TL;DR

This paper proves that learning an optimal path graphical model from data is NP-hard across multiple scoring approaches, despite the problem being a restricted form of the polynomially solvable optimal tree model.

Contribution

It establishes the computational hardness of finding optimal path models, extending the understanding of complexity in graphical model learning.

Findings

NP-hardness of learning optimal path graphical models for ML, MDL, and Bayesian methods

Contrasts with polynomial solvability of optimal tree models

Highlights computational challenges in certain graphical model structures

Abstract

I consider the problem of learning an optimal path graphical model from data and show the problem to be NP-hard for the maximum likelihood and minimum description length approaches and a Bayesian approach. This hardness result holds despite the fact that the problem is a restriction of the polynomially solvable problem of finding the optimal tree graphical model.