In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (2008). By analyzing $n$-dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with $n$ integer variables is a $t$-branch split cut. We use this result to give a finite cutting-plane algorithm to solve mixed-integer programs. We also show that the minimum value $t$, for which all facets of polyhedral mixed-integer sets with $n$ integer variables can be generated as $t$-branch split cuts, grows exponentially with $n$. In particular, when $n=3$, we observe that not all facet-defining inequalities are 6-branch split cuts.
IBM Research Report