Before revision:  We then consider general (non-quadratic) functions. We give a broad parameterized family of monotone functions which include submodular functions and the just-discussed supermodular family of discrete quadratics. Such set functions are called \gamma-meta-submodular. We develop local search algorithms with approximation factors that depend only on \gamma. We show that the \gamma-meta-submodular families include well-known function classes including meta-submodular functions (\gamma=0), proportionally submodular (\gamma= 1 ), and diversity functions based on negative-type distances or Jensen-Shannon divergence (both \gamma=2) and (semi-)metric diversity)-approximation, independent of n.
After revision:  For general one-sided smooth functions, we show the continuous process gives an \Omega( 1 ), and diversity functions based on negative-type distances or Jensen-Shannon divergence (both \gamma=2) and (semi-)metric diversity)-approximation, independent of n.