Computing the Stability Number of a Graph Via Linear and Semidefinite Programming

We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines de Klerk and Pasechnik’s approach to approximating 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.