  We study the problem of finding fair and efficient allocations of a set of
indivisible items to a set of agents, where each item may be a good (positively
valued) for some agents and a bad (negatively valued) for others, i.e., a mixed
manna. As fairness notions, we consider arguably the strongest possible
relaxations of envy-freeness and proportionality, namely envy-free up to any
item (EFX and EFX$_0$), and proportional up to the maximin good or any bad
(PropMX and PropMX$_0$). Our efficiency notion is Pareto-optimality (PO).
  We study two types of instances:
  (i) Separable, where the item set can be partitioned into goods and bads, and
(ii) Restricted mixed goods (RMG), where for each item $j$, every agent has
either a non-positive value for $j$, or values $j$ at the same $v_j>0$. We
obtain polynomial-time algorithms for the following:
  (i) Separable instances: PropMX$_0$ allocation.
  (ii) RMG instances: Let pure bads be the set of items that everyone values
negatively.
  - PropMX allocation for general pure bads.
  - EFX+PropMX allocation for identically-ordered pure bads.
  - EFX+PropMX+PO allocation for identical pure bads.
  Finally, if the RMG instances are further restricted to binary mixed goods
where all the $v_j$'s are the same, we strengthen the results to guarantee
EFX$_0$ and PropMX$_0$ respectively.
