A Parameterized Family of Meta-Submodular Functions

TL;DR

This paper introduces a broad family of functions called ta-submodular, unifying submodular and supermodular functions, and develops local search algorithms with approximation guarantees depending on a parameter mma.

Contribution

It defines the ta-submodular family encompassing various known functions and provides approximation algorithms with guarantees based on mma.

Findings

Includes well-known classes like meta-submodular and diversity functions.

Provides local search algorithms with mma-dependent approximation factors.

Unifies submodular and supermodular functions under a common framework.

Abstract

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 ($\gamma = \sigma$).