Before revision:  Together with the adapted continuous greedy this leads to a O(\sigma ^{3/2})-approximation . This is the best asymptotic approximation known for this class of quadratics and the evidence suggests that it may be tight.
After revision:  Together with the adapted continuous greedy this leads to a O(\sigma ^{3/2})-approximation for maximizing a \sigma-semi-metric diversity function subject to matroid constraint. This improves upon the previous best bound of \Omega(1/\sigma^2) and we give evidence that it may be tight.