On the p-media polytope of special class of graphs

Dans cet article nous étudions les inégalités de cycles impairs. Nous donnons un nouvel algorithme de séparation de ces inégalités, et nous montrons que lorsque nous ajoutons ces inégalités aux inégalités de la relaxation linéaire nous obtenons le polytope du p-médian dans la classe des graphes sans Y. Pour cela nous caractérisons d'abord les points extrêmesfractionnaires du système défini par la relaxation linéaire dans les graphes sans Y.

On the p-media polytope of special class of graphs

In this paper we consider a well known class of valid inequalities for the p-median and the uncapacitated facility location polytopes, the odd cycle inequalities. It is known that their separation problem is polynomially solvable. We give a new polynomial separation algorithm based on a reduction from the original graph. Then, we define a nontrivial class of graphs, where the odd cycle inequalities together with the linear relaxations of both the p-median and uncapacitated facility location problems, suffice to describe the associated polytope. To do this, we first give a complete description of the fractional extreme points of the linear relaxation for the p-median polytope in that class of graphs.Dans cet article nous étudions les inégalités de cycles impairs. Nous donnons un nouvel algorithme de séparation de ces inégalités, et nous montrons que lorsque nous ajoutons ces inégalités aux inégalités de la relaxation linéaire nous obtenons le polytope du p-médian dans la classe des graphes sans Y. Pour cela nous caractérisons d'abord les points extrêmesfractionnaires du système défini par la relaxation linéaire dans les graphes sans Y

Clique-circulants and the stable set polytope of fuzzy circular interval graphs

In this paper, we give a complete and explicit description of the rank facets of the stable set polytope for a class of claw-free graphs, recently introduced by Chudnovsky and Seymour (Proceedings of the Bristish Combinatorial Conference, 2005), called fuzzy circular interval graphs. The result builds upon the characterization of minimal rank facets for claw-free graphs by Galluccio and Sassano (J. Combinatorial Theory 69:1-38, 2005) and upon the introduction of a superclass of circulant graphs that are called clique-circulants. The new class of graphs is invetigated in depth. We characterize which clique-circulants C are facet producing, i.e. are such that Sigma upsilon epsilon V(C) chi(upsilon) <= alpha(C) is a facet of STAB(C), thus extending a result of Trotter (Discrete Math. 12:373-388, 1975) for circulants. We show that a simple clique family inequality (Oriolo, Discrete Appl. Math. 132(2):185-201, 2004) may be associated with each clique-circulant C subset of G, when G is fuzzy circular interval. We show that these inequalities provide all the rank facets of STAB(G), if G is a fuzzy circular interval graph. Moreover we conjecture that, in this case, they also provide all the non-rank facets of STAB(G) and offer evidences for this conjecture