The interval order polytope of a digraph

We introduce the interval order polytope of a digraph D as the convex hull of interval order inducing arc subsets of D. Two general schemes for producing valid inequalities are presented. These schemes have been used implicitly for several polytopes and they are applied here to the interval order polytope. It is shown that almost all known classes of valid inequalities of the linear ordering polytope can be explained by the two classes derived from these schemes. We provide two applications of the interval order polytope to combinatorial optimization problems for which to our knowledge no polyhedral descriptions have been given so far. One of them is related to analyzing DNA subsequences.