We consider the participatory budgeting problem where each of $n$ voters specifies additive utilities over $m$ candidate projects with given sizes, and the goal is to choose a subset of projects (i.e., a committee) with total size at most $k$. Participatory budgeting mathematically generalizes multiwinner elections, and both have received great attention in computational social choice recently. A well-studied notion of group fairness in this setting is core stability: Each voter is assigned an "entitlement" of $\frac{k}{n}$, so that a subset $S$ of voters can pay for a committee of size at most $|S| \cdot \frac{k}{n}$. A given committee is in the core if no subset of voters can pay for another committee that provides each of them strictly larger utility. This provides proportional representation to all voters in a strong sense. In this paper, we study the following auditing question: Given a committee computed by some preference aggregation method, how close is it to the core? Concretely, how much does the entitlement of each voter need to be scaled down by, so that the core property subsequently holds? As our main contribution, we present computational hardness results for this problem, as well as a logarithmic approximation algorithm via linear program rounding. We show that our analysis is tight against the linear programming bound. Additionally, we consider two related notions of group fairness that have similar audit properties. The first is Lindahl priceability, which audits the closeness of a committee to a market clearing solution. We show that this is related to the linear programming relaxation of auditing the core, leading to efficient exact and approximation algorithms for auditing. The second is a novel weakening of the core that we term the sub-core, and we present computational results for auditing this notion as well.

翻译：我们研究参与式预算问题：其中$n$位选民对$m$个给定规模的候选项目指定可加效用，目标是选择总规模不超过$k$的项目子集（即委员会）。参与式预算在数学上推广了多胜者选举，两者近期在计算社会选择领域均受到广泛关注。该场景下一个被深入研究的群体公平性概念是核心稳定性：每位选民被分配$\frac{k}{n}$的“权益”，使得选民子集$S$能够支付规模不超过$|S| \cdot \frac{k}{n}$的委员会。若不存在选民子集能支付另一个为其中每位成员提供严格更高效用的委员会，则给定委员会处于核心中。这以强形式为所有选民提供了比例代表性。本文研究以下审计问题：对于通过某种偏好聚合方法计算出的委员会，其与核心的接近程度如何？具体而言，需要将每位选民的权益按多大比例缩减，才能随后满足核心性质？作为主要贡献，我们给出了该问题的计算复杂性结果，以及通过线性规划舍入实现的对数近似算法。我们证明该分析与线性规划下界是紧的。此外，我们考察了两种具有相似审计特性的群体公平性概念。其一是林达尔可定价性，用于审计委员会与市场出清解的接近程度。我们证明该概念与核心审计的线性规划松弛相关，从而推导出高效的精确与近似审计算法。其二是我们称为次核心的核心概念新弱化形式，我们也给出了审计该概念的计算结果。