Exact Graph Learning via Integer Programming
Lucas Kook, Søren Wengel Mogensen
TL;DR
GLIP presents an exact, nonparametric graph-learning framework that casts conditional-independence based structure learning as a mixed-integer linear program, enabling globally optimal recovery of DAGs, ADMGs, DMGs and chain graphs. A key innovation is a minimal-length encoding of graphical separations that dramatically reduces the number of variables while preserving correctness, allowing exact learning on larger graphs than prior exact methods. The approach yields representations of Markov (or weak) equivalence classes and provides provable guarantees of optimality, with an Open R implementation in glip. Empirically, GLIP demonstrates strong performance relative to exact ASP methods and competitive results against state-of-the-art approximate methods across simulated and benchmark data, while offering practical warmstart strategies for larger problems. The work broadens exact causal discovery to richer graph classes and offers practical tools for nonparametric causal inference in complex systems.
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 as causal discovery if the graphs are given a causal interpretation. 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. As a result, 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 nonparametric conditional independence testing and integer programming. We reformulate the graph learning problem as an integer-programming problem and prove that solving the 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 implementation in the openly available R package 'glip' which supports learning (acyclic) directed (mixed) graphs and chain graphs. From the resulting output one can compute representations of the corresponding Markov equivalence classes or weak equivalence classes. Empirically, we demonstrate that our approach is faster than other existing exact graph learning procedures for a large fraction of instances and graphs of various sizes. GLIP also achieves state-of-the-art performance on simulated data and benchmark datasets across all aforementioned classes of graphs.