On mixed-integer sets with two integer variables

Abstract

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 recently in [3]). We then extend this observation to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables provided that 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 about the equivalence of different definitions of split cuts presented in Cook, Kannan, and Schrijver [4]. This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard [6]. 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.

On the relative strength of different generalizations of split cuts

“…However, these two dominance relationships are not known to be strict. In [21] the authors show that there is a crooked cross cut that is not implied by a single cross cut; however, this result does not rule out the possibility that the cross closure (which potentially contains infinitely many cuts) is always equal to the crooked cross closure. In this paper we establish that 3-branch split cuts strictly dominate crooked cross cuts which, in turn, strictly dominate 2-branch split cuts.…”

“…However, Theorem 1.2 establishes that this is not the case, as it was proved in [21,Theorem 3.1] that (1) equals the crooked cross closure of the mixed-integer set.…”

“…Subsequently, it was shown in [19] that t-branch split cuts strictly dominate k-branch split cuts for all t > k > 0. In addition, it is known that 3-branch split cuts dominate crooked cross cuts which, in turn, dominate cross cuts [21,20]. However, these two dominance relationships are not known to be strict.…”

“…These cuts were first studied by Li and Richard [18] and recently Dash and Günlük extended some of their results [19]. Dash, Dey and Günlük [20,21] study 2-branch split cuts (and call them cross cuts) and crooked cross cuts (which are derived using three linearly dependent split disjunctions). Crooked cross cuts subsume cross cuts and are implied by 3-branch split cuts.…”

Computational Experiments with Cross and Crooked Cross Cuts

“…Further, it is shown in Dash et al (2011) that P CC ⊆ P C , but the containment is not strict (Dash et al 2012b). …”