Facets of the linear ordering polytope: A unification for the fence family through weighted graphs

The binary choice polytope appeared in the investigation of the binary choice problem formulated by Guilbaud and Block and Marschak. It is nowadays known to be the same as the linear ordering polytope from operations research (as studied by Grötschel, Jünger and Reinelt).The central problem is to find facet-defining linear inequalities for the polytope. Fence inequalities constitute a prominent class of such inequalities (Cohen and Falmagne; Grötschel, Jünger and Reinelt). Two different generalizations exist for this class: the reinforced fence inequalities of Leung and Lee, and independently Suck, and the stability-critical fence inequalities of Koppen. Together with the fence inequalities, these inequalities form the fence family. Building on previous work on the biorder polytope by Christophe, Doignon and Fiorini, we provide a new class of inequalities which unifies all inequalities from the fence family. The proof is based on a projection of polytopes. The new class of facet-defining inequalities is related to a specific class of weighted graphs, whose definition relies on a curious extension of the stability number. We investigate this class of weighted graphs which generalize the stability-critical graphs.