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

Abstract

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 [Discrete Optim., 9 (2012), pp. 209--215] extending results in Andersen, Cornuejols, and Li [Math. Program, 102 (2005), pp. 457--493]. 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 bounded max-facet-width polyhedra as building blocks, and show that any family of $t$-branch sets where each set is the union of $t$ polyhedral sets that have bounded max-facet-width has a fin...