(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 EFX0), and proportional up to the maximin good or any bad (PropMX and PropMX0). 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 vj>0. We obtain polynomial-time algorithms for the following: (i) Separable instances: PropMX0 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 vj's are the same, we strengthen the results to guarantee EFX0 and PropMX0 respectively.