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We study the problem of finding fair and efficient allocations of a set ofindivisible items to a set of agents, where each item may be a good (positivelyvalued) for some agents and a bad (negatively valued) for others, i.e., a mixedmanna. As fairness notions, we consider arguably the strongest possiblerelaxations of envy-freeness and proportionality, namely envy-free up to anyitem (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 haseither a non-positive value for $j$, or values $j$ at the same $v_j>0$. Weobtain 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 valuesnegatively.  - 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 goodswhere all the $v_j$'s are the same, we strengthen the results to guaranteeEFX$_0$ and PropMX$_0$ respectively.

(Almost) Envy-Free, Proportional and Efficient Allocations of an Indivisible Mixed Manna