Dual Formulation of the Chordal Graph Conjecture

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.