  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.
