  We study almost-envy-freeness in house allocation, where $m$ houses are to be
allocated among $n$ agents so that every agent receives exactly one house. An
envy-free allocation need not exist, and therefore we may have to settle for
relaxations of envy-freeness. But typical relaxations such as envy-free up to
one good do not make sense for house allocation, as every agent is required to
receive exactly one house. Hence we turn to different aggregate measures of
envy as markers of fairness. In particular, we define the amount of envy
experienced by an agent $a$ w.r.t. an allocation to be the number of agents
that agent $a$ envies under that allocation. We quantify the envy generated by
an allocation using three different metrics: 1) the number of agents who are
envious; 2) the maximum amount of envy experienced by any agent; and 3) the
total amount of envy experienced by all agents, and look for allocations that
minimize one of the three metrics. We thus study three computational problems
corresponding to each of the three metrics and prove a host of algorithmic and
hardness results. We also suggest practical approaches for these problems via
integer linear program (ILP) formulations and report the findings of our
experimental evaluation of ILPs. Finally, we study the price of fairness (PoF),
which quantifies the loss of welfare we must suffer due to the fairness
requirements, and we prove a number of results on PoF, including tight bounds
as well as algorithms that simultaneously optimize both welfare and fairness.
