  Fair division has emerged as a very hot topic in multiagent systems, and
envy-freeness is among the most compelling fairness concepts. An allocation of
indivisible items to agents is envy-free if no agent prefers the bundle of any
other agent to his own in terms of value. As envy-freeness is rarely a feasible
goal, there is a recent focus on relaxations of its definition. An approach in
this direction is to complement allocations with payments (or subsidies) to the
agents. A feasible goal then is to achieve envy-freeness in terms of the total
value an agent gets from the allocation and the subsidies.
  We consider the natural optimization problem of computing allocations that
are {\em envy-freeable} using the minimum amount of subsidies. As the problem
is NP-hard, we focus on the design of approximation algorithms. On the positive
side, we present an algorithm that, for a constant number of agents,
approximates the minimum amount of subsidies within any required accuracy, at
the expense of a graceful increase in the running time. On the negative side,
we show that, for a super-constant number of agents, the problem of minimizing
subsidies for envy-freeness is not only hard to compute exactly (as a folklore
argument shows) but also, more importantly, hard to approximate.
