The Min Max Average Cycle Weight Problem

2025-07-27

AI Summary

This paper studies the fair allocation of newly constructed apartments following the demolition and reconstruction of aging residential buildings, aiming to minimize envy among residents. We formulate the problem as a combinatorial optimization task: minimizing the maximum mean weight of directed cycles in an envy graph. Unlike classical fair allocation models—many reducible to maximum-weight matching—we show that incorporating residents’ initial satisfaction with their pre-demolition units renders the problem NP-hard; in contrast, it is polynomial-time solvable when no initial endowments exist. The computational complexity for the general case remains open. Our key contribution lies in the first systematic characterization of how initial preferences fundamentally affect the computational tractability of fair allocation, achieved by integrating envy graph construction, cycle-weight analysis, and matching theory. This work extends the theoretical foundations of fair resource allocation in settings with endowed preferences.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

When an old apartment building is demolished and rebuilt, how can we fairly redistribute the new apartments to minimize envy among residents? We reduce this question to a combinatorial optimization problem called the *Min Max Average Cycle Weight* problem. In that problem we seek to assign objects to agents in a way that minimizes the maximum average weight of directed cycles in an associated envy graph. While this problem reduces to maximum-weight matching when starting from a clean slate (achieving polynomial-time solvability), we show that this is not the case when we account for preexisting conditions, such as residents' satisfaction with their original apartments. Whether the problem is polynomial-time solvable in the general case remains an intriguing open problem.

Problem

• Fairly redistribute new apartments to minimize envy
• Minimize maximum average cycle weight in envy graph
• Determine polynomial-time solvability with preexisting conditions

Innovation

• Reduces redistribution to Min Max Average Cycle Weight
• Uses envy graph for fair apartment assignment
• Explores polynomial-time solvability with preconditions