Steiner trees and Polyhedra

this paper, we study polyhedra STP(G,S) and CON(G,S). We describe a new class of facet defining inequalities for the STP(G,S) that generalizes the families of constraints so called Steiner partition inequalities and odd hole inequalities introduced by Chopra and Rao [3]. We show that these inequalities may define facets for 2-trees, which invalidates a conjecture of Chopra and Rao [4]. We also discuss the closely related Steiner connected subgraph polytope. We describe some procedures of construction of facets from facets for CON(G,S). Using this, we obtain a complete description of both CON(G,S) and STP(G,S) for a special case of series-parallel graphs. Computational applications are also discussed. 2 Valid inequalitie

On the Polytope of the Vertex Separator Problem

Given G = (V, E) a connected undirected graph and a positive integer β(|V |), the vertex separator problem is to find a partition of V into three nonempty subsets A, B, C such that (i) there is no edge between the nodes of A and those of B, (ii) max{|A|, |B|} ≤ β(|V |) and (iii) |C| is minimum. In this paper, we consider the problem from a polyhedral point of view. We first propose a new integer programming formulation for the problem. Then we provide several valid inequalities for the polytope which generalize those introduced by Balas and De Souza [1], and give conditions under which these inequalities define facets

On k-edge-connected Polyhedra: Box-TDIness in Series-Parallel Graphs

Given a connected graph G=(V,E) and an integer (formula presented), the connected graph H=(V,F) where F is a family of elements of E, is a k-edge-connected spanning subgraph of G if H remains connected after the removal of any k-1 edges. The convex hull of the k-edge-connected spanning subgraphs of a graph G forms the k-edge-connected spanning subgraph polyhedron of G. We prove that this polyhedron is box-totally dual integral if and only if G is series-parallel. In this case, we also provide an integer box-totally dual integral system describing this polyhedron