Concave Aspects of Submodular Functions

TL;DR

This paper explores the relationship between submodular functions and concavity, characterizing super-differentials and optimality conditions for maximization, enhancing understanding of their mathematical structure.

Contribution

It provides a detailed characterization of super-differentials and polyhedra related to submodular functions, clarifying their connection to concavity and optimization.

Findings

Characterization of super-differentials for submodular functions

Optimality conditions for submodular maximization

Insights into the concave aspects of submodular functions

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].