The dual polyhedron to the chordal graph polytope and the rebuttal of the chordal graph conjecture

Studený, Milan; Cussens, James; Kratochvíl, Václav

doi:10.1016/j.ijar.2021.07.014

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…16. With this we give a partial answer to a question by Studený, Cussens and Kratochvíl, [16]. We conjecture that the chordal graph polytope fulfills the rhombus criterion, Conjecture 3.4, and we raise the question whether it is true for the entire imset polytope CIM n .…”

Section: Introductionmentioning

confidence: 71%

“…Primarily we show that almost all pairs of vertices of the chordal graph polytope fulfill the rhombus criterion, Theorem 3. 16. With this we give a partial answer to a question by Studený, Cussens and Kratochvíl, [16].…”

Section: Introductionmentioning

confidence: 74%

“…Thus the chordal graph polytope is spanned by the subset of all vertices of the characteristic imset polytope that does not contain v-structures. This polytope is well studied [19,16] from the perspective of causal discovery and describing certain facets. Studený, Cussens, and Kratochvíl raised the question of describing the edges of CGP n , and to this end we conjecture that the answer is in the rhombus criterion.…”

Section: Characteristic Imset Polytopesmentioning

confidence: 99%

“…It would be desirable to give an explicit description of when c G , c H are neighbors in CGP n for split graphs G, H. From the long proof of Theorem 3. 16 we can deduce that it happens (1) if there exists a unique…”

Section: Case III Kmentioning

confidence: 99%

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Greedy Causal Discovery Is Geometric

Linusson¹,

Restadh²,

Solus³

2023

SIAM J. Discrete Math.

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The purpose of this paper is twofold. We investigate a simple necessary condition, called the rhombus criterion, for two vertices in a polytope not to form an edge and show that in many examples of 0/1-polytopes it is also sufficient. We explain how also when this is not the case, the criterion can give a good algorithm for determining the edges of high-dimenional polytopes.In particular we study the Chordal graph polytope, which arises in the theory of causality and is an important example of a characteristic imset polytope. We prove that, asymptotically, for almost all pairs of vertices the rhombus criterion holds. We conjecture it to hold for all pairs of vertices.

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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 74%

Section: Characteristic Imset Polytopesmentioning

confidence: 99%

Section: Case III Kmentioning

confidence: 99%

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Greedy Causal Discovery Is Geometric

Linusson¹,

Restadh²,

Solus³

2023

SIAM J. Discrete Math.

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“…Due to its relevance in the problem of causal discovery, the polyhedral geometry of CIM n has been studied. In [22,23,34] families of facets of CIM n and certain subpolytopes of CIM n are identified. Similar results have also been found for families of edges in [14,15].…”

Section: Preliminariesmentioning

confidence: 99%

Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

Hollering¹,

Johnson²,

Portakal³

et al. 2022

Preprint

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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 propose directions for future work.

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