Reforming an Envy-Free Matching
This addresses incremental improvements in matching algorithms for resource allocation problems, with implications for domains like school choice or job assignments.
The paper tackles the problem of reforming an envy-free matching by exchanging items to reach a unique reformist envy-free matching, proving it can be found in polynomial time, but finding the shortest sequence is computationally hard in some cases while polynomial-time algorithms exist for others.
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.