  In this paper we initiate the study of finding fair and efficient allocations
of an indivisible mixed manna: Divide m indivisible items among n agents under
the fairness notion of maximin share (MMS) and the efficiency notion of Pareto
optimality (PO). A mixed manna allows an item to be a good for some agents and
a chore for others. The problem of finding $\alpha$-MMS allocation for the
(near) best $\alpha\in(0,1]$ for which it exists, remains unresolved even for a
goods manna with constantly many agents, while the problem of finding
$\alpha$-MMS+PO allocation is unexplored for any $\alpha\in(0,1]$.
  We make significant progress on the above questions for a mixed manna. First,
we show that for any $\alpha>0$, an $\alpha$-MMS allocation may not always
exist, thus ruling out solving the problem for a fixed $\alpha$. Second,
towards computing $\alpha$-MMS+PO allocation for the best possible $\alpha$, we
obtain a dichotomous result: We derive two conditions and show that the problem
is tractable under these two conditions, while dropping either renders the
problem intractable. The two conditions are: (i) number of agents is a
constant, and (ii) for every agent, her absolute value for all the items is at
least a constant factor of her total (absolute) value for all the goods or all
the chores.
  In particular, first, for instances satisfying (i) and (ii) we design a PTAS
- an efficient algorithm to find an $(\alpha-\epsilon)$-MMS and $\gamma$-PO
allocation when given $\epsilon,\gamma>0$, for the highest possible
$\alpha\in(0,1]$. Second, we show that if either condition is not satisfied
then finding an $\alpha$-MMS allocation for any $\alpha\in(0,1]$ is NP-hard,
even when a solution exists for $\alpha=1$. To the best of our knowledge, ours
is the first algorithm that ensures both approximate MMS and PO guarantees.
