We study fair division of indivisible chores among $n$ agents with additive disutility functions. Two well-studied fairness notions for indivisible items are envy-freeness up to one/any item (EF1/EFX) and the standard notion of economic efficiency is Pareto optimality (PO). There is a noticeable gap between the results known for both EF1 and EFX in the goods and chores settings. The case of chores turns out to be much more challenging. We reduce this gap by providing slightly relaxed versions of the known results on goods for the chores setting. Interestingly, our algorithms run in polynomial time, unlike their analogous versions in the goods setting. We introduce the concept of $k$ surplus which means that up to $k$ more chores are allocated to the agents and each of them is a copy of an original chore. We present a polynomial-time algorithm which gives EF1 and PO allocations with $(n-1)$ surplus. We relax the notion of EFX slightly and define tEFX which requires that the envy from agent $i$ to agent $j$ is removed upon the transfer of any chore from the $i$'s bundle to $j$'s bundle. We give a polynomial-time algorithm that in the chores case for $3$ agents returns an allocation which is either proportional or tEFX. Note that proportionality is a very strong criterion in the case of indivisible items, and hence both notions we guarantee are desirable.

翻译：我们研究在具有可加负效用的 $n$ 个智能体之间公平分配不可分杂务的问题。对于不可分物品，两个被广泛研究的公平性概念是至多一个/任意一个物品的无嫉妒性（EF1/EFX），而经济效率的标准概念是帕累托最优性（PO）。在商品和杂务场景中，关于EF1和EFX的已知结果之间存在显著差距。杂务情形被证明更具挑战性。我们通过提供商品场景中已知结果在杂务场景下的略微宽松版本来缩小这一差距。有趣的是，与商品场景中的对应版本不同，我们的算法在多项式时间内运行。我们引入了 $k$ 多余物品的概念，这意味着最多额外分配 $k$ 个杂务给智能体，且每个杂务都是原始杂务的副本。我们提出了一种多项式时间算法，能够在具有 $(n-1)$ 个多余物品的情况下给出EF1和PO分配。我们略微放宽了EFX的概念，定义了tEFX，它要求当任意杂务从智能体 $i$ 的集合转移到智能体 $j$ 的集合后，智能体 $i$ 对智能体 $j$ 的嫉妒消失。我们给出了一种多项式时间算法，在 $3$ 个智能体的杂务场景中，该算法返回的分配要么是成比例的，要么是tEFX的。注意，在不可分物品的情况下，比例性是一个非常强的标准，因此我们保证的这两个概念都是令人满意的。