Exact Graph Learning via Integer Programming

TL;DR

This paper introduces an exact, globally optimal graph learning method using integer programming, capable of handling larger graphs efficiently and outperforming existing approaches in accuracy and speed.

Contribution

The authors develop a nonparametric graph learning framework reformulated as a mixed-integer program, enabling exact recovery of complex graphs with improved scalability and performance.

Findings

Method often faster than existing exact procedures.

Achieves state-of-the-art performance on simulated data.

Supports learning various graph types including directed and chain graphs.

Abstract

Learning the dependence structure among variables in complex systems is a central problem across medical, natural, and social sciences. These structures can be naturally represented by graphs, and the task of inferring such graphs from data is known as graph learning or causal discovery. Existing approaches typically rely on restrictive assumptions about the data-generating process, employ greedy oracle algorithms, or solve approximate formulations of the graph learning problem. Therefore, they are either sensitive to violations of central assumptions or fail to guarantee globally optimal solutions. We address these limitations by introducing a nonparametric graph learning framework based on conditional independence testing and integer programming. We reformulate the graph learning problem as a mixed-integer program and prove that solving this integer-programming problem provides a globally optimal solution to the original graph learning problem. Our method leverages efficient encodings of graphical separation criteria, enabling the exact recovery of larger graphs than was previously feasible. We provide an open-source R package 'glip' which supports learning (acyclic) directed (mixed) graphs and chain graphs. We demonstrate that our approach is often faster than existing exact graph learning procedures and achieves state-of-the-art performance on simulated and benchmark data across all aforementioned classes of graphs.