Abstract
The idea of an integer linear programming approach to structural learning of decomposable graphical models led to the study of the so-called chordal graph polytope. An open mathematical question is what is the minimal set of linear inequalities defining this polytope. Some time ago we came up with a specific conjecture that the polytope is defined by so-called clutter inequalities. In this theoretical paper we give a dual formulation of the conjecture. Specifically, we introduce a certain dual polyhedron defined by trivial equality constraints, simple monotonicity inequalities and certain inequalities assigned to incomplete chordal graphs. The main result is that the list of (all) vertices of this bounded polyhedron gives rise to the list of (all) facet-defining inequalities of the chordal graph polytope. The original conjecture is then equivalent to a statement that all vertices of the dual polyhedron are zero-one vectors. This dual formulation of the conjecture offers a more intuitive view on the problem and allows us to disprove the conjecture.
Original language | English |
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Title of host publication | Proceedings of Machine Learning Research |
Subtitle of host publication | Proceedings of the 10th International Conference on Probabilistic Graphical Models |
Editors | Manfred Jaeger, Thomas D. Nielsen |
Pages | 449-460 |
Number of pages | 12 |
Volume | 138 |
Publication status | Published - 25 Sept 2020 |
Event | 10th International Conference on Probabilistic Graphical Models - Hotel Comwell Rebild Bakker, Skørping, Denmark Duration: 23 Sept 2020 → 25 Sept 2020 https://pgm2020.cs.aau.dk/ |
Publication series
Name | Proceedings of Machine Learning Research |
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Volume | 138 |
ISSN (Electronic) | 2640-3498 |
Conference
Conference | 10th International Conference on Probabilistic Graphical Models |
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Abbreviated title | PGM 2020 |
Country/Territory | Denmark |
City | Skørping |
Period | 23/09/20 → 25/09/20 |
Internet address |
Bibliographical note
Funding Information:The research of Milan Studený and Václav Kratochvíl has been supported by the grant GAČR number 19-04579S.