Optimizing a Generalized Gini Index in Stable Marriage Problems: NP-Hardness, Approximation and a Polynomial Time Special Case

TL;DR

This paper explores fairness in stable marriage problems using a Generalized Gini Index, proving NP-hardness, and providing approximation and special case algorithms for optimizing fairness criteria.

Contribution

It introduces the first complexity analysis for GGI-based fairness in stable marriage problems and offers efficient algorithms for specific cases.

Findings

Determined GGI optimization is NP-hard in stable marriage.

Developed a polynomial-time 2-approximation algorithm.

Provided an exact polynomial-time algorithm for fixed non-zero weights.

Abstract

This paper deals with fairness in stable marriage problems. The idea studied here is to achieve fairness thanks to a Generalized Gini Index (GGI), a well-known criterion in inequality measurement, that includes both the egalitarian and utilitarian criteria as special cases. We show that determining a stable marriage optimizing a GGI criterion of agents' disutilities is an NP-hard problem. We then provide a polynomial time 2-approximation algorithm in the general case, as well as an exact algorithm which is polynomial time in the case of a constant number of non-zero weights parametrizing the GGI criterion.