Approximation algorithms for k-submodular maximization under fairness constraints and size constraints

Abstract

Submodular optimization is an important part of combinatorial optimization, and meanwhile, it has been widely used in the fields of economy, computer science, etc. In recent years, many generalizations of the submodular maximization problem have been proposed, such as the k-submodular maximization problem, which considers not only whether the elements of the ground set are selected, but also which set of k-set they are selected into. Many machine learning problems, including influence maximization with k kinds of topics and sensor placement with k kinds of sensors, can be modeled as k-submodular maximization models. In practical applications, elements usually have certain attributes, such as gender, age, race, and so on. Based on these attributes we can group elements in the ground set. For fairness, we want the number of elements in each group to be relatively balanced, which introduces group fairness. Group fairness can be conceptualized in various ways, but typically falls into two primary categories: equity-fairness and equality-fairness. At present, the existing algorithms on k-submodular do not consider group fairness constraints, which can result in the number of elements in some groups being too high or too low. This motivates us to study k-submodular maximization under group fairness constraints. In this paper, we consider intersection constraints of two different categories of group fairness constraints and total size constraint, respectively. For the k-submodular maximization problem under equity-fairness constraints and total size constraint, we design an offline approximation algorithm with an approximate ratio of 1/2-ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/2-\epsilon $$\end{document} for the case of monotone, and an approximate ratio of 1/3-ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/3-\epsilon $$\end{document} for non-monotone. For the k-submodular maximization problem under equality-fairness constraints and total size constraint, we get an approximate ratio of 1/2 for the case of monotone and an approximate ratio of 0.041 for the case of non-monotone.