On the Holt-Klee Property for Oriented Matroid Programming

Abstract

The Holt-Klee theorem says that the graph of a d-polytope, with edges oriented by a linear function on P that is not constant on any edge, admits d independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank d + 1 on n + 2 elements, which include those from d-polytopes with n facets, admit d independent monotone paths from source to sink if d ≤ 4. This was previously only known to hold for d ≤ 3 and n ≤ 6.