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.

翻译：暂无翻译