On the integral dicycle packings and covers and the linear ordering polytope

Abstract

The linear ordering polytope P LO n 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 P LO n , there corresponds a digraph inducing it. Let D be a digraph that induces a facet-defining inequality for P LO n , 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. Dridi's description of P LO 5 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 Wagner's decomposition, we obtain that τ = v in K 3.3-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.