Facets of linear signed order polytopes

• Authors: Samuel Fiorini;Peter Fishburn
• Venue: Discrete Applied Mathematics
• Year: 2003

Self-reflecting signed orders have been proposed to aid assessment of preferences between subsets of an n-item set {1,2,... ,n} by considering desirabilities of excluding as well as including items in a set. A linear signed order for n is a linear order on the 2n-element set {1,...,n} U {1*,...,n*}, where (x*)* = x, which satisfies the self-reflection property x y ⇔ y* x*. The linear signed order polytope Qn for n is defined in a standard way as a polytope in [0,1]2n(2n-1). It has dimension n2. We note a complete equation system for Qn and specify all facet defining inequalities for n ≤ 4. Additional classes of facets for larger n that are not induced by a lifting lemma are identified. Comparisons to linear ordering polytopes are included.