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Toric ideals of characteristic imsets via quasi-independence gluing

Benjamin Hollering, Joseph Johnson, Irem Portakal and Liam Solus

Vol. 14 (2023), No. 2, 109–131
Abstract

Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper, we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and provide direction for future work.

Keywords
characteristic imset, quasi-independence, toric fiber product, polytope
Mathematical Subject Classification
Primary: 62E10, 62H22, 62D20, 62R01, 13P25, 13P10
Milestones
Received: 5 September 2022
Revised: 12 September 2023
Accepted: 28 September 2023
Published: 16 May 2024
Authors
Benjamin Hollering
Technische Universität München
Munich
Germany
Joseph Johnson
Institutionen för Matematik
KTH Royal Institute of Technology
Stockholm
Sweden
Irem Portakal
Technische Universität München
Munich
Germany
Liam Solus
Institutionen för Matematik
KTH Royal Institute of Technology
Stockholm
Sweden