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.
翻译:我们研究的是,在一组代理商中找到公平和高效地分配一组不可分割的物品的问题,其中每个物品可能对某些物剂是好的(积极估价的),对另一些物剂是坏的(消极估价的),即混杂的曼纳。作为公平概念,我们认为,可以认为无妒忌和相称性方面可能最有力的放松,即对任何物品(EFX和EFX$0美元)而言,无嫉妒,与美元或任何坏的成比例(PropMX和PropMX$0美元);我们的效率概念是Pareto-最佳(PO)。 我们研究两种情况:(i) 配方,该物品可以分成货物和坏的商品,以及(ii) 限制的混合商品(RMG),对于每件物品,每个物价都是非正的美元,或每件的RV_j>0美元。我们进一步获得以下的多时算法:(i) Probalto-Problexi) 例:纯的分配情况:纯MAX最后,每件的正分配。