The Complexity of Minimizing Envy in House Allocation

The house allocation problem asks for m houses to be allocated among n agents so that every agent receives exactly one house. The preferences of the agents over houses can be modeled as weak orders, of which two special cases are strict rankings and binary valuations. Given an allocation ϕ of houses to agents, an agent a envies agent b if a receives a house ϕ(a) that they like strictly less than ϕ(b), the house allocated to b. The amount of envy experienced by an agent a is the number of agents b that it envies with respect to ϕ. We consider the computational problem of finding allocations that minimize: the number of agents who are envious, the maximum envy experienced by any agent, or the total amount of envy experienced by all agents. We investigate the complexity of all three optimization objectives for both strict rankings as well as binary valuations. We show that these problems are FPT when parameterized by the number of houses and the number of agents. When parameterized by solution size, i.e, the value of the optimization objective, we demonstrate W-hardness in the first objective and para-NP-hardness for the last objective. We also consider these questions in the setting of restricted domains and also suggest practical approaches for these problems via ILP formulations.