A Generalization of the Perfect Graph Theorem Under the Disjunctive Index

Abstract

In this paper we relate antiblocker duality between polyhedra, graph theory and the disjunctive procedure. In particular, we analyze the behavior of the disjunctive procedure over the clique relaxation, K(G), of the stable set polytope in a graph G and the one associated to its complementary graph, K(Ḡ). We obtain a generalization of the Perfect Graph Theorem proving that the disjunctive indices of K(G) and K(Ḡ) always coincide.

On the behavior of the N+-operator under blocker duality

“…Nevertheless, since we are dealing with convex sets in [0, 1] n instead of cones, we restate the definitions in the following way (cf. [1,10]). Definition 2.1.…”

Semidefinite Programming and Integer Programming

“…One crucial issue in duality theory consists in identifying sufficient conditions that insure equality in (2), also called Strong Duality. The following condition insures strong duality.…”

“…Given a graph G = (V, E), the polytope QSTAB(G), consisting of the vectors x ∈ R V + satisfying the clique inequalities (42), is a linear relaxation of the stable set polytope STAB(G), stronger than the fractional stable set polytope FRAC(G) considered earlier in Section 3.5. Aguilera, Escalante and Nasini [2] show that the rank of the polytope QSTAB(G) with respect to the Balas-Ceria-Cornuéjols procedure is equal to the rank of QSTAB(G), where G is the complementary graph of G.…”

Clutter nonidealness

“…In this way Aguilera et al defined in [2] an imperfection index of a graph G through the behavior of the disjunctive operator [3] on QSTAB(G). In other direction, Gerke and McDiarmid defined in [9] an imperfection index from the quotient between the fractional chromatic number and the clique number in a weighted graph.…”