  Submodular function maximization has found a wealth of new applications in
machine learning models during the past years. The related supermodular
maximization models (submodular minimization) also offer an abundance of
applications, but they appeared to be highly intractable even under simple
cardinality constraints. Hence, while there are well-developed tools for
maximizing a submodular function subject to a matroid constraint, there is much
less work on the corresponding supermodular maximization problems.
  We give a broad parameterized family of monotone functions which includes
submodular functions and a class of supermodular functions containing diversity
functions. Functions in this parameterized family are called
\emph{$\gamma$-meta-submodular}. We develop local search algorithms with
approximation factors that depend only on the parameter $\gamma$. We show that
the $\gamma$-meta-submodular families include well-known classes of functions
such as meta-submodular functions ($\gamma=0$), metric diversity functions and
proportionally submodular functions (both with $\gamma=1$), diversity functions
based on negative-type distances or Jensen-Shannon divergence (both with
$\gamma=2$), and $\sigma$-semi metric diversity functions ($\gamma = \sigma$).
