On combinatorial properties of linear program digraphs [chapter]

David Avis, Sonoko Moriyama
2009 CRM Proceedings and Lecture notes AMS  
The possible pivot operations of the simplex method to solve a linear program can be represented as a directed graph defined on the skeleton of the feasible region P . We consider the case that P is bounded, i.e., a convex polytope. The directed graph is called an LP digraph. LP digraphs are known to satisfy the following three properties: acyclicity, unique sink orientation(USO), and the Holt-Klee property. The three properties are not generally sufficient for a directed graph on the skeleton
more » ... f P to be an LP digraph. In this paper, we first survey some previous results on LP digraphs, showing relationships among the three properties. Then we introduce a new necessary property for a directed graph on the skeleton of P to be an LP digraph, called the shelling property. We analyze the relationships between the shelling property and the three existing properties, showing that it is stronger than a combination of acyclicity and USO for non-simple polytopes in dimension at least four. In all other cases it is equivalent to the intersection of these two properties.
doi:10.1090/crmp/048/01 fatcat:gqzkpxfccfgg3k65ref5dp4mxa