无嫉妒房屋分配中补贴最小化的复杂度
The Complexity of Minimizing Subsidies in Envy-Free House Allocation
AI总结：
本文研究无嫉妒房屋分配中最小化补贴的问题，证明二元实例下总补贴界为(n-1)且紧，一般及二元效用下该问题NP难，有限类型二元效用下可多项式求解，还提出两类一般效用智能体的最小补贴多项式算法。
AI中文摘要：
房屋分配问题是经典的单侧匹配问题，涉及根据偏好将m套房屋分配给n个智能体，每个智能体恰好分配一套房屋。该场景中研究的各类目标里，无嫉妒性是最广泛采用的公平性准则之一。由于无嫉妒房屋分配并非总能存在，我们引入补贴来应对这一挑战，旨在计算出能实现无嫉妒性且总补贴最小的分配方案。对于二元实例，我们证明总补贴最多为(n-1)即可保证房屋分配的无嫉妒性，且该界是紧的。基于一般效用下已知的NP难性，我们进一步证明，即使在二元效用下，计算使总补贴最小的分配方案也是NP难的。然而，当具有二元效用的智能体类型数量有限时，该问题可在多项式时间内解决。最后，我们提出一种多项式时间算法，用于计算具有一般效用的两类智能体实现无嫉妒性所需的最小补贴。
英文摘要：
The house allocation problem is a classical one-sided matching problem that concerns the assignment of a set of $m$ houses to $n$ agents according to their preferences, where each agent is assigned exactly one house. Among the various objectives studied in this setting, envy-freeness is one of the most widely adopted fairness criteria. As envy-free house allocations do not always exist, we address this challenge by introducing subsidies and aim to compute allocations that achieve envy-freeness with minimum total subsidy. For binary instances, we show that a total subsidy of at most $(n-1)$ suffices to guarantee envy-freeness in house allocation, and this bound is tight. Building on the known NP-hardness for general utilities, we further show that computing an allocation that minimizes the total subsidy is NP-hard, even under binary utilities. However, when there are only a bounded number of types of agents with binary utilities, the problem can be solved in polynomial time. Finally, we present a polynomial time algorithm that computes the minimum subsidy required to achieve envy-freeness for two types of agents with general utilities.
