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. Dridi's 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 Wagner's 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.
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Let Dn be the complete digraph on n nodes. Let P~.o denote the convex hull of all the incidence vectors of arc sets of linear orderings of the nodes of D. (i.e. these are exactly the incidence vectors of the acyclic tournaments of Dn). P~.o is called the linear ordering polytope. The linear ordering problem consists of maximizing a linear function over P~.o. The investigation of the linear ordering problem is motivated by *Corresponding author. tpart of this work was done as part of the author's M.Sc. thesis, done under the supervision of Dov Monderer and Michal Penn, in the Department of Applied Mathematics. Technion, Haifa, Israel. "Research of this author was partially supported by the Mendence France Fellowship Trust.