Fair Submodular Maximization over a Knapsack Constraint

TL;DR

This paper studies fair submodular maximization under a knapsack constraint, providing polynomial-time algorithms with constant approximation for fixed colors and tight expected ratios when relaxing constraints.

Contribution

It introduces algorithms achieving constant and near-optimal approximation ratios for fair submodular maximization with knapsack constraints under various conditions.

Findings

Polynomial-time algorithm with constant approximation for fixed number of colors.

Expected satisfaction relaxation yields a tight approximation ratio of (1-1/e-ε).

Progress on a challenging open problem in fair submodular maximization.

Abstract

We consider fairness in submodular maximization subject to a knapsack constraint, a fundamental problem with various applications in economics, machine learning, and data mining. In the model, we are given a set of ground elements, each associated with a weight and a color, and a monotone submodular function defined over them. The goal is to maximize the submodular function while guaranteeing that the total weight does not exceed a specified budget (the knapsack constraint) and that the number of elements selected for each color falls within a designated range (the fairness constraint). While there exists some recent literature on this topic, the existence of a non-trivial approximation for the problem -- without relaxing either the knapsack or fairness constraints -- remains a challenging open question. This paper makes progress in this direction. We demonstrate that when the number of colors is constant, there exists a polynomial-time algorithm that achieves a constant approximation with high probability. Additionally, we show that if either the knapsack or fairness constraint is relaxed only to require expected satisfaction, a tight approximation ratio of $(1-1/e-\epsilon)$ can be obtained in expectation for any $\epsilon >0$.