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

We study certain semidefinite and linear 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 show that the exact value of the stability number α(G) is attained by the semidefinite approximation of order α(G)− 1. We also give a closed-form expression for the values computed by the linear programming approximations. The closed-form expression readily reveals 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.