Undominated complexes of cut polytopes

Abstract

Polytopes are basic geometric objects of fundamental importance for Linear Programming (i.e., optimization of linear functions under linear constraints). One of the most important research topic of polytope theory is its combinatorial aspect: the combinatorial structure of polytopes. This thesis is primarily concerned with the concept of the undominated set of a polytope, i.e., the set of all those points of a polytope such that there is no other different point in the polytope which is component-wise less than or equal to it. The interest in undominated sets originates in the fact that two polytopes (in the first orthant) have the same dominant if and only if their undominated sets coincide. The dominant of a polyhedron is the geometric object that is of interest when minimizing linear objective functions with nonnegative coe cients only. For such minimization problems over a polyhedron, the initial polyhedron can be replaced by another polyhedron with the same dominant but which is easier to describe, e.g. with fewer inequalities. It turns out that the undominated set of a polytope is a polyhedral complex (the undominated complex), formed by those faces that are also faces of the dominant, or, equivalently, the bounded faces of the dominant. One result we establish about the undominated complexes of general polytopes is that they are contractible. While their topological structure thus is rather simple, we provide some three-dimensional examples showing that their geometry can nevertheless look surprisingly complicated. The main part of the thesis is devoted to investigating the undominated complexes of the polytopes associated with the cuts in the complete undirected graph on nodes {1, 2, ... , n} that separate at least one of the nodes 1, 2, ... , Xi from node n. We provide characterizations of the combinatorial structures of the undominated complexes of those cut polytopes for Xi = 1, 2, 3, showing that those complexes are pure simplicial complexes of dimension n - 2, n - 1, n in these cases, respectively. We also provide a conjecture for the combinatorial structures of those complexes for general , for which a partial prove is given by the main contribution.