Georgia Institute of Technology College of Sciences

In the past decade, the catalog of algorithms available to combinatorial optimizers has been substantially extended to settings which allow submodular objective functions. One significant recent result was a tight (1-1/e)-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint. These algorithmic developments were happening concurrently with research that found a wealth of new applications for submodular optimization in machine learning, recommender systems, and algorithmic game theory.

The related supermodular maximization models also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints and even when the function has a nice structure. For example, the densest subgraph problem - suspected to be highly intractable - can be expressed as a monotonic supermodular function which has a particularly nice form. Namely, the objective can be expressed by a quadratic form $x^T A x$ where $A$ is a non-negative, symmetric, 0-diagonal matrix. On the other hand, when the entries $A(u,v)$ form a metric, it has been shown that the associated maximization problems have constant factor approximations. Inspired by this, we introduce a parameterized class of non-negative functions called meta-submodular functions that can be approximately maximized within a constant factor. This class includes metric diversity, monotone submodular and other objectives appearing in the machine learning and optimization literature. A general meta-submodular function is neither submodular nor supermodular and so its multi-linear extension does not have the nice convexity/concavity properties which hold for submodular functions. They do, however, have an intrinsic one-sided smoothness property which is essential for our algorithms. This smoothness property might be of independent interest.