On the Polyhedrality of Closures of Multibranch Split Sets and Other Polyhedra with Bounded Max-Facet-Width

For a fixed integer t > 0, we say that a t-branch split set (the union of t split sets) is dominated by another one on a polyhedron P if all cuts for P obtained from the first t-branch split set are implied by cuts obtained from the second one. We prove that given a rational polyhedron P , any arbitrary family of t-branch split sets has a finite subfamily such that each element of the family is dominated on P by an element from the subfamily. The result for t = 1 (i.e., for split sets) was proved by Averkov (2012) extending results in Andersen, Cornuéjols and Li (2005). Our result implies that the closure of P with respect to any family of t-branch split sets is a polyhedron. We extend this result by replacing split sets with polyhedral sets with bounded max-facet-width as building blocks and show that any family of such sets also has a finite dominating subfamily. This result generalizes a result of Averkov (2012) on bounded max-facet-width polyhedra.