  We consider the multiwinner election problem where the goal is to choose a
committee of $k$ candidates given the voters' utility functions. We allow
arbitrary additional constraints on the chosen committee, and the utilities of
voters to belong to a very general class of set functions called $\beta$-self
bounding. When $\beta=1$, this class includes XOS (and hence, submodular and
additive) utilities. We define a novel generalization of core stability called
restrained core to handle constraints and consider multiplicative
approximations on the utility under this notion.
  Our main result is the following: If a smooth version of Nash Welfare is
globally optimized over committees within the constraints, the resulting
committee lies in the $e^{\beta}$-approximate restrained core for $\beta$-self
bounding utilities and arbitrary constraints. As a result, we obtain the first
constant approximation for stability with arbitrary additional constraints even
for additive utilities (factor of $e$), and the first analysis of the stability
of Nash Welfare with XOS functions even with no constraints.
  We complement this positive result by showing that the $c$-approximate
restrained core can be empty for $c<16/15$ even for approval utilities and one
additional constraint. Furthermore, the exponential dependence on $\beta$ in
the approximation is unavoidable for $\beta$-self bounding functions even with
no constraints.
  We next present improved and tight approximation results for simpler classes
of utility functions and simpler types of constraints. We also present an
extension of restrained core to extended justified representation with
constraints and show an existence result for matroid constraints. We finally
generalize our results to the setting with arbitrary-size candidates and no
additional constraints. Our techniques are different from previous analyses and
are of independent interest.
