on the Integral Dicycle Packings and Covers and the Linear ordering Polytope

The linear ordering polytope P~.o is defined as the convex hull of all the incidence vectors of the acyclic tournaments on n nodes, it is known that for every facet of P~.o there corresponds a digraph inducing it. Let D be a digraph that induces a facet-defining inequality for P~.o, 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 ~ of a minimum inteoral dicycle cover is greater than the value 3" of a minimum dicycle cover. We show that 3" 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 dieycle packing. We prove that if D is a certain digraph with a two-node cut satisfying ~ = v in each part, then ~ = v in D as well. Dridi's description of P~o enables a simple derivation oftbe fact that ~ = v for any digraph on 5 nodes. Combining these results with the theorem of Luccbesi and Younger for planar digraphs as well as Wagner's decomposition, we obtain that ~ = v in K3.3-frce digraphs. This last result was proved recently by Barahona et al. (1990) using polyhedral techniques while our proof is based mainly on combinatorial tools.