A full description of polytopes related to the index of the lowest nonzero row of an assignment matrix

Walid Ben-Ameur (1, 2, 3) , Antoine Glorieux (1, 2, 3) , José Neto (1, 2, 3)

1 METHODES-SAMOVAR - Méthodes et modèles pour les réseaux
2 TSP - RS2M - Département Réseaux et Services Multimédia Mobiles
3 CNRS - Centre National de la Recherche Scientifique

Consider a {0, 1} assignment matrix where each column contains exactly one coefficient equal to 1 and let h be the index of the lowest row that is not identically equal to the zero row. We give a full description of the convex hull of all feasible assignments appended with the extra parameter h. This poly-tope and some of its variants naturally appear in the context of several combi-natorial optimization problems including frequency assignment, job scheduling, graph orientation, maximum clique, etc. We also show that the underlying separation problems are solvable in polynomial time and thus optimization over those polytopes can be done in polynomial time.

Mots clés: Combinatorial optimization problem; Polytope; Polyhedra; Polyhedron full description; Hyperplane representation; Assignment matrix; Separation problem; Polynomial separation; Branch & cut algorithm; Cutting plane algorithm; Frequency assignment; Polynomial time; Maximum clique

Domaines: Combinatoire [math.CO]; Informatique [cs]; Mathématique discrète [cs.DM]

Walid Ben-Ameur, Antoine Glorieux, José Neto. A full description of polytopes related to the index of the lowest nonzero row of an assignment matrix. ISCO 2016 : 4th International Symposium on Combinatorial Optimization, May 2016, Vietri Sul Mare, Italy. pp.13-25, ⟨10.1007/978-3-319-45587-7_2⟩. ⟨hal-01497880⟩