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

0545447 - ÚTIA 2022 RIV US eng J - Journal Article
Studený, Milan - Cussens, J. - Kratochvíl, Václav
The dual polyhedron to the chordal graph polytope and the rebuttal of the chordal graph conjecture.
International Journal of Approximate Reasoning. Roč. 138, č. 1 (2021), s. 188-203. ISSN 0888-613X. E-ISSN 1873-4731
R&D Projects: GA ČR(CZ) GA19-04579S
Institutional support: RVO:67985556
Keywords : learning decomposable models * chordal graph polytope * clutter inequalities * dual polyhedron
OECD category: Pure mathematics
Impact factor: 4.452, year: 2021
Method of publishing: Limited access
http://library.utia.cas.cz/separaty/2021/MTR/studeny-0545447.pdf https://www.sciencedirect.com/science/article/pii/S0888613X21001316?via%3Dihub

The integer linear programming approach to structural learning of decomposable graphical models led us earlier to the concept of a chordal graph polytope. An open mathematical question motivated by this research is what is the minimal set of linear inequalities defining this polytope, i.e. what are its facet-defining inequalities, and we came up in 2016 with a specific conjecture that it is the collection of so-called clutter inequalities. In this theoretical paper we give an implicit characterization of the minimal set of inequalities. Specifically, we introduce a dual polyhedron (to the chordal graph polytope) defined by trivial equality constraints, simple monotonicity inequalities and certain inequalities assigned to incomplete chordal graphs. Our main result is that the vertices of this polyhedron yield the facet-defining inequalities for the chordal graph polytope. We also show that the original conjecture is equivalent to the condition that all vertices of the dual polyhedron are zero-one vectors. Using that result we disprove the original conjecture: we find a vector in the dual polyhedron which is not in the convex hull of zero-one vectors from the dual polyhedron.
Permanent Link: http://hdl.handle.net/11104/0322204