  We study fair division of indivisible goods in a single-parameter
environment. In particular, we develop truthful social welfare maximizing
mechanisms for fairly allocating indivisible goods. Our fairness guarantees are
in terms of solution concepts which are tailored to address allocation of
indivisible goods and, hence, provide an appropriate framework for fair
division of goods. This work specifically considers fairness in terms of envy
freeness up to one good (EF1), maximin share guarantee (MMS), and Nash social
welfare (NSW).
  Our first result shows that (in a single-parameter environment) the problem
of maximizing welfare, subject to the constraint that the allocation of the
indivisible goods is EF1, admits a polynomial-time, 1/2-approximate, truthful
auction. We further prove that this problem is NP-Hard and, hence, an
approximation is warranted. This hardness result also complements prior works
which show that an arbitrary EF1 allocation can be computed efficiently.
  We also establish a bi-criteria approximation guarantee for the problem of
maximizing social welfare under MMS constraints. In particular, we develop a
truthful auction which efficiently finds an allocation wherein each agent gets
a bundle of value at least $\left(1/2 - \varepsilon \right)$ times her maximin
share and the welfare of the computed allocation is at least the optimal, here
$\varepsilon >0$ is a fixed constant. We complement this result by showing that
maximizing welfare is computationally hard even if one aims to only satisfy the
MMS constraint approximately.
