Approximating the sign rank of a matrix
I believe this is an open question.  

Lee and Schraibman in "An approximation algorithm for approximation rank" show that the approximation rank can be approximated to a constant factor by a polynomial time algorithm.  To do so, they relate a semidefinite programming quantity $\gamma_2^\alpha$ to the approximation rank, where the $\alpha$ is some finite parameter greater than 1.  Taking $\alpha$ to the limit of infinity yields the sign-rank but their result does not give anything in this setting.