Fair Submodular Maximization over a Knapsack Constraint

TL;DR

This work addresses fair submodular maximization under a knapsack constraint (FKSM), formalizing a problem with a monotone submodular objective and per-group fairness bounds. It introduces two key techniques—knapsack truncating to reduce FKSM to a tractable $\overline{F}KSM$ instance and randomized weighted pipage rounding to preserve knapsack feasibility and fairness in expectation while maintaining submodular value—then combines them with continuous greedy to obtain strong guarantees. The main results show a constant-factor approximation when the number of color groups is fixed (via knapsack truncation) and a tight $(1-\frac{1}{e}-\epsilon)$-approximation in expectation when fairness is relaxed to hold in expectation (via weighted pipage rounding). These contributions advance fair optimization under strict resource and equity constraints and open questions for non-constant group counts and more general constraint structures.

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 $(1-1/e-ε)$ can be obtained in expectation for any $ε>0$.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Fair Knapsack Polytope
• Definition 2.2: Multilinear Extension
• Lemma 2.3: aamas/PKL21
• Lemma 2.4
• Theorem 3.1
• Lemma 3.1
• proof
• Lemma 3.2: chekuri2009