Derivations of large classes of facet-defining inequalities of the weak order polytope using ranking structures

TL;DR

This paper introduces new classes of facet-defining inequalities for the weak order polytope by leveraging ranking structures with ties, unifying and extending previous enumeration-based results.

Contribution

It derives novel FDIs for the weak order polytope using ranking representations, connecting them to existing enumeration-based inequalities.

Findings

New classes of FDIs derived from ranking structures.

Existing FDIs shown to be special cases of the new classes.

Enhanced understanding of the weak order polytope's facets.

Abstract

The study of ordering polytopes has been essential to the solution of various challenging combinatorial optimization problems. For instance, the incorporation of facet defining inequalities (FDIs) from these polytopes in branch-and-cut approaches represents among the most effective solution methodologies known to date for some of these problems. The weak order polytope, defined as the convex hull of the characteristic vectors of all binary orders on $n$ alternatives that are reflexive, transitive, and complete, has been particularly important for tackling problems in computational social choice, preference aggregation, and comparative probability. For the most part, FDIs for the weak order polytope have been obtained through enumeration and through derivation from FDIs of other combinatorial polytopes. This paper derives new classes of FDIs for the weak order polytope by utilizing the equivalent representation of a weak order as a ranking of $n$ objects that allows ties and by grouping characteristic vectors that share certain ranking structures. Furthermore, we demonstrate that a number of FDIs previously obtained through enumeration are actually special cases of these ranking-based FDIs.