0, 1/2-Cuts and the Linear Ordering Problem: Surfaces That Define Facets

We find new facet-defining inequalities for the linear ordering polytope generalizing the well-known Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1 2 }-cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the projective plane. In that sense, Möbius ladder inequalities have the same ‘shape’ as the projective plane. Inspired by the classification of surfaces, a classic result in topology, we prove that a surface has facet-defining {0, 1 2 }-cuts of the same ‘shape’ if and only if it is nonorientable.