A comparison between lift-and-project indices and imperfection ratio on web graphs

In this paper we study the lift-and-project polyhedral operators defined by Lovász and Schrijver [L. Lovász, A. Schrijver, Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optimization 1, (1991), pp. 166–190] and by Balas, Ceria and Cornuéjols [3] on the clique relaxation of the stable set polytope of webs. We prove they have the same perfomance when starting from the clique relaxation of the family of webs W s ( k + 1 ) + k k . Considering the lift-and-project strength of facets for the stable set polytope on webs defined in [S. Bianchi, M. Escalante, M.S. Montelar, Strength of facets for the set covering and set packing polyhedra on circulant matrices, Electronic Notes in Discrete Mathematics 35, (2009), pp. 109-114], we obtain that the facets of maximum strength for the family W s ( k + 1 ) + k k are also the facets of maximum strength according to Goemansʹ measure [M. Goemans, Worst-case Comparison of Valid Inequalities for the TSP, Mathematical Programming, Fifth Integer Programming and Combinatorial Optimization Conference, LNCS 1084, Vancouver, Canada, (1996), pp. 415–429]. This last result is obtained by means of the imperfection index and imperfection ratio defined in [N. Aguilera, G. Nasini, M. Escalante, A generalization of the perfect graph theorem under the disjunctive index, Mathematics of Operations Research Vol. 27, No 3, (2002) pp. 460–469] and [S. Gerke, C. McDiarmid, Graph imperfection, Journal of Combinatorial Theory Series B 83, (2001), pp. 58–78].