On the integral dicycle packings and covers and the linear ordering polytope Original Research Article

The linear ordering polytope PLOn is defined as the convex hull of the incidence vectors of the acyclic tournaments on n nodes. It is known that for every facet of PLOn, there corresponds a digraph inducing it. Let D be a digraph that induces a facet-defining inequality for PLOn, that is nonequivalent to a trivial inequality or to a 3-dicycle inequality. We show that for such a digraph the following holds: the value τ of a minimum integral dicycle cover is greater than the value τ∗ of a minimum dicycle cover. We show that τ∗ can be found by minimizing a linear function over a polytope which is defined by a polynomial number of constraints. Let v denote the value of a maximum integral dicycle packing. We prove that if D is a certain digraph with a two-node cut satisfying τ = v in each part, then τ = v in D as well. Dridiʹs description of PLO5 enables a simple derivation of the fact that τ = v for any digraph on 5 nodes. Combining these results with the theorem of Lucchesi and Younger for planar digraphs as well as Wagnerʹs decomposition, we obtain that τ = v in K3.3-free digraphs. This last result was proved recently by Barahona et al. (1990) using polyhedral techniques while our proof is based mainly on combinatorial tools.