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arXiv · 2006.16784

Concave Aspects of Submodular Functions

Rishabh Iyer·Jeff Bilmes
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Abstract

Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial-time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP-hard, admits constant-factor approximation guarantees, and concave functions composed with modular functions are submodular. In this paper, we try to provide a more complete picture of the relationship between submodularity with concavity. We characterize the super-differentials and polyhedra associated with upper bounds and provide optimality conditions for submodular maximization using the-super differentials. This paper is a concise and shorter version of our longer preprint [3].

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Rishabh Iyer, Jeff Bilmes. 2020-06-27. Concave Aspects of Submodular Functions. https://arxiv.org/abs/2006.16784

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