The vertex separator problem: a polyhedral investigation

• Published in: Mathematical programming Vol. 103; no. 3; pp. 583 - 608
• Main Authors: Balas, Egon, Souza, Cid C. de
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.07.2005; Springer Nature B.V
• Subjects: Applied sciences; Exact sciences and technology; Grants; Integer programming; Investigations; Linear algebra; Mathematical programming; Operational research and scientific management; Operational research. Management science; Studies; Polyhedron; Branch and cut method; Convex polytope; Connected graph; Mixed integer programming; Vertex(graph); Separator; Non directed graph
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-005-0574-7

The vertex separator (VS) problem in a graph G =( V , E ) asks for a partition of V into nonempty subsets A , B , C such that there is no edge between A and B , and | C | is minimized subject to a bound on max{| A |,| B |}. We give a mixed integer programming formulation of the problem and investigate the vertex separator polytope (VSP), the convex hull of incidence vectors of vertex separators. Necessary and sufficient conditions are given for the VSP to be full dimensional. Central to our investigation is the relationship between separators and dominators. Several classes of valid inequalities are investigated, along with the conditions under which they are facet defining for the VSP. Some of our proofs combine in new ways projection with lifting. [PUBLICATION ABSTRACT