Before revision:  This captures metric diversity maximization ( \sigma =2) and negative-type diversity (\sigma=4) . We also develop new methods (inspired by swap rounding and approximate integer decomposition)for rounding quadratics over a matroid polytope. Together with the adapted continuous greedy this leads to a O(\sigma ^{3/2})-approximation .
After revision:  This captures metric diversity maximization ( \sigma ^{3/2})-approximation .