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

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 nonpositive value for j, or values j at the same v j > 0. We obtain polynomial-time algorithms for the following: • Separable instances: PropMX 0 allocation. • 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.