Computing the stability number of a graph via linear and semidefinite programming Javier Pe˜ na * Tepper School of Business Carnegie Mellon University Pittsburgh, PA 15213-3890, USA e-mail: jfp@andrew.cmu.edu Juan Vera † Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213, USA email: jvera@andrew.cmu.edu Luis F. Zuluaga ‡ Faculty of Administration University of New Brunswick PO Box 4400, Fredericton NB E38 5X9, CANADA e-mail: lzuluaga@unb.ca April 27, 2006 Abstract We study certain linear and semidefinite programming lifting approxi- mation schemes for computing the stability number of a graph. Our work is based on, and refines de Klerk and Pasechnik’s approach to approximat- ing the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875–892). We provide a closed-form expression for the values computed by the linear programming approximations. We also show that the exact value of the stability number α(G) is attained by the semidefinite approximation of order α(G) - 1 as long as α(G) ≤ 6. Our results reveal some sharp differences between the linear and the semidefinite approximations. For instance, the value of the linear programming approximation of any order is strictly larger than α(G) whenever α(G) > 1. * Supported by NSF grant CCF-0092655. † Partially supported by NSF grant CCF-0092655. ‡ Supported by NSERC grant 31814-05 and FDF, University of New Brunswick. 1