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

TL;DR

This work develops a theoretical framework for maximin (Rawlsian) fairness in offline and dynamic combinatorial optimization, defining static and dynamic fair objectives for value and proportion. It proves general bounds linking SF and DF fairness to standard optimal objectives and proposes a baseline Separate-$\mathcal{A}$-Solve method to construct fair distributions over solutions. The Max-Cut running example reveals tight gaps between dynamic fairness and static fairness under edge and node utilities, showing both fundamental limitations and algorithmic avenues, including LP-based exact DF-MP computation and limitations of Goemans-Williamson for fairness. The study highlights the practical relevance of fairness over time and advocates further exploration of iterative and quantum-assisted approaches to generate fair solution distributions in complex combinatorial settings.

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.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Theorem 3: monotonicity of subproblems
• Corollary 4
• Theorem 7
• Proposition 10
• Proposition 11
• Corollary 12
• Proposition 13
• Proposition 14