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 ($\gamma = \sigma$).

中文翻译：

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元子模块函数的参数化族

在过去的几年里，子模块函数最大化在机器学习模型中发现了大量的新应用。相关的超模最大化模型（子模最小化）也提供了丰富的应用，但即使在简单的基数约束下，它们似乎也非常难以处理。因此，虽然存在用于最大化受拟阵约束的子模函数的成熟工具，但在相应的超模最大化问题上的工作要少得多。我们给出了一个广泛的参数化单调函数族，其中包括子模函数和一类包含多样性函数的超模函数。此参数化系列中的函数称为 \emph{$\gamma$-meta-submodular}。我们使用仅依赖于参数 $\gamma$ 的近似因子开发局部搜索算法。我们表明 $\gamma$-meta-submodular 系列包括众所周知的函数类别，例如元子模函数（$\gamma=0$）、度量多样性函数和成比例的子模函数（均具有 $\gamma=1 $)、基于负型距离或 Jensen-Shannon 散度的多样性函数（均具有 $\gamma=2$）和 $\sigma$-半度量多样性函数（$\gamma = \sigma$）。