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