A Note on Split Rank of Intersection Cuts

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that dlog2(l)e is a lower bound on the split rank of the intersection cut, where l is the number of integer points lying on the boundary of the restricted lattice-free set satisfying the condition that no two points lie on the same facet of the restricted lattice-free set. The use of this result is illustrated to obtain a lower bound of dlog2(n + 1)e on the split rank of n-row mixing inequalities. Over the years, many classes of cuts have been proposed for solving unstructured mixed integer programs that can be used within the branch-and-cut framework; see Nemhauser and Wolsey [29], Marchand, Martin, Weismantel and Wolsey [27] and Johnson, Nemhauser and Savelsbergh [25]. Among the many classes of cutting planes proposed, Split cuts (Balas [5]) which are equivalent to the Gomory Mixed Integer cuts (Gomory [21]) and the Mixed Integer Rounding inequalities (Nemhauser and Wolsey [30]) form one of the most successful classes of cutting planes used to solve general mixed integer programs; see for example Balas et al. [7] and Bixby and Rothberg [11]. It is therefore natural to compare other classes of valid cutting planes with split cuts. One possible method of comparison is to ask if a recursive application of split cuts will generate the target class of inequalities. If the recursive application of split cuts does generate a target inequality, the minimum number of steps necessary to obtain the target inequality gives a measure of the efficacy of a cutting-plane-algorithm that uses only split cuts. These questions are related to the question of determining the split rank of the inequality. While the exact split rank for a general class of inequalities may be difficult to obtain, bounds on the split rank are more easily obtainable. A finite upper bound on the split rank indicates that the inequality under study can be obtained by recursively applying split cuts. On the other hand, a lower bound on the split rank indicates how difficult it is to obtain an inequality using split cuts. Therefore, if the lower bound on the split rank of an inequality is high, it may be better to apply this inequality directly to the LP relaxation of the problem instead of generating it using a sequence of split cuts. In this note, we study the split rank of valid inequalities for the following class of problems,