Nash Welfare Guarantees for Fair and Efficient Coverage

TL;DR

This paper introduces a polynomial-time approximation algorithm for maximizing Nash social welfare in coverage problems, balancing fairness and efficiency, and proves the problem's APX-hardness.

Contribution

It presents the first polynomial-time approximation algorithm with a constant factor for Nash social welfare maximization in coverage problems and establishes its computational hardness.

Findings

Achieves a (18 + o(1))-approximation ratio.

Applicable to instances with an FPTAS for weight maximization.

Proves Nash social welfare maximization is APX-hard.

Abstract

We study coverage problems in which, for a set of agents and a given threshold $T$, the goal is to select $T$ subsets (of the agents) that, while satisfying combinatorial constraints, achieve fair and efficient coverage among the agents. In this setting, the valuation of each agent is equated to the number of selected subsets that contain it, plus one. The current work utilizes the Nash social welfare function to quantify the extent of fairness and collective efficiency. We develop a polynomial-time $\left(18 + o(1) \right)$-approximation algorithm for maximizing Nash social welfare in coverage instances. Our algorithm applies to all instances wherein, for the underlying combinatorial constraints, there exists an FPTAS for weight maximization. We complement the algorithmic result by proving that Nash social welfare maximization is APX-hard in coverage instances.