Proportionality (PROP) is one of the simplest and most intuitive fairness criteria used for allocating items among agents with additive utilities. However, when the items are indivisible, ensuring PROP becomes unattainable, leading to increased focus on its relaxations. In this paper, we focus on the relaxation of proportionality up to any item (PROPX), where proportionality is satisfied if an arbitrary item is removed from every agent's allocation. We show that PROPX is an appealing fairness notion for the allocation of indivisible chores, which approximately implies some share-based notions, such as maximin share (MMS) and AnyPrice share (APS). We further provide a comprehensive understanding of PROPX allocations, regarding the computation, approximation, and compatibility with efficiency. On top of these, we extend the study to scenarios where agents do not share equal liability towards the chores, and approximate PROPX allocations using partial information about agents' utilities.

中文翻译：

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不可分割的杂务几乎按比例分配：计算、近似和效率

比例（PROP）是最简单、最直观的公平标准之一，用于在具有附加效用的代理之间分配项目。然而，当这些项目不可分割时，确保 PROP 就变得遥不可及，导致人们更加关注其放松。在本文中，我们重点关注对任何项目（PROPX）的比例放宽，如果从每个代理的分配中删除任意项目，则满足比例。我们表明，PROPX 对于分配不可分割的杂务来说是一个有吸引力的公平概念，这大约意味着一些基于份额的概念，例如最大最小份额（MMS）和 AnyPrice 份额（APS）。我们进一步提供了对 PROPX 分配的全面理解，包括计算、近似以及与效率的兼容性。除此之外，我们将研究扩展到代理人不承担同等家务责任的场景，并使用有关代理人公用事业的部分信息来近似 PROPX 分配。