On the Polyhedrality of Closures of Multibranch Split Sets and Other Polyhedra with Bounded Max-Facet-Width
证明了对于任意一族t分支分裂集，存在有限子族使得每个元素都被该子族中的某个元素在多面体P上支配，从而推出P关于该族的闭包是多面体；并将此结果推广到有界最大面宽多面体构成的t分支集。
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, Cornuéjols, 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 finite dominating subfamily with respect to $P$. This latter result generalizes a result of Averkov [Discrete Optim., 9 (2012), pp. 209--215] on bounded max-facet-width polyhedra (corresponding to the case $t=1$).