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

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985556%3A_____%2F21%3A00545447" target="_blank" >RIV/67985556:_____/21:00545447 - isvavai.cz</a>
Result on the web
<a href="https://www.sciencedirect.com/science/article/pii/S0888613X21001316?via%3Dihub" target="_blank" >https://www.sciencedirect.com/science/article/pii/S0888613X21001316?via%3Dihub</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.ijar.2021.07.014" target="_blank" >10.1016/j.ijar.2021.07.014</a>

Result language
angličtina
Original language name
The dual polyhedron to the chordal graph polytope and the rebuttal of the chordal graph conjecture
Original language description
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.
Czech name
—
Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA19-04579S" target="_blank" >GA19-04579S: Conditonal independence structures: methods of polyhedral geometry</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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