Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

TL;DR

This paper explores the algebraic structure of characteristic imsets related to causal graphs, introducing a new method called quasi-independence gluing to compute Gr"obner bases, with applications to trees and cycles.

Contribution

It introduces quasi-independence gluing, a novel generalization of toric fiber product, for computing Gr"obner bases of toric ideals of characteristic imsets.

Findings

Successfully computed Gr"obner bases for tree-related faces of the polytope.

Proposed a new technique applicable to cycle-related ideals.

Outlined future directions for algebraic analysis of characteristic ideals.

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 Gr\"obner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gr\"obner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work.