Expected Maximin Fairness in Max-Cut and other Combinatorial Optimization Problems

TL;DR

This paper investigates the concept of maximin fairness in combinatorial optimization problems, highlighting theoretical bounds and demonstrating challenges through the Max-Cut problem.

Contribution

It provides theoretical insights into maximin fairness bounds and explores the complexities of implementing fairness in combinatorial problems like Max-Cut.

Findings

Optimal maximin solutions are bounded by non-maximin solutions.

Stochastic maximin solutions outperform deterministic ones in expectation.

Challenges in defining and applying maximin fairness in Max-Cut.

Abstract

Maximin fairness is the ideal that the worst-off group (or individual) should be treated as well as possible. Literature on maximin fairness in various decision-making settings has grown in recent years, but theoretical results are sparse. In this paper, we explore the challenges inherent to maximin fairness in combinatorial optimization. We begin by showing that (1) optimal maximin-fair solutions are bounded by non-maximin-fair optimal solutions, and (2) stochastic maximin-fair solutions exceed their deterministic counterparts in expectation for a broad class of combinatorial optimization problems. In the remainder of the paper, we use the special case of Max-Cut to demonstrate challenges in defining and implementing maximin fairness.