On mixed-integer sets with two integer variables

We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alternative characterization of the crooked cross cut closure of mixed-integer sets similar to the one on the equivalence of different definitions of split cuts presented in Cook, Kannan, and Schrijver (1990). This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard (2008). Finally, we extend our results to mixed-integer sets that are defined as the set of points (with some components being integral) inside a closed, bounded, convex set.