We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

翻译：当每个代理被分配到一个单一的项目时,我们考虑改革一个无嫉妒的匹配问题。根据一个无嫉妒的匹配,我们考虑将一个代理的项目与该代理所偏爱的未分配项目交换,从而产生另一个无嫉妒的匹配。我们尽可能多地重复这一操作。我们证明,由此产生的无嫉妒的匹配的独特性取决于最初的无嫉妒的匹配的选择,并且可以在多民族时间中找到。我们称其结果匹配一个改革派的无嫉妒的匹配,然后我们研究一个最短的序列,以便从最初的无嫉妒的匹配中获得改革派的无嫉妒匹配。我们证明,即使每个代理接受最多四个项目,而每个项目被最多三个代理所接受,也很难计算出一个最短的序列。另一方面,当每个代理接受最多三个项目或每个项目被最多两个代理所接受时,我们给出了多元时间算法。关于不适应性和固定参数(可选取性)也得到了讨论。