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.

翻译：我们考虑多赢家选举问题，目标是在选民效用函数下选择包含$k$名候选人的委员会。我们允许对所选委员会施加任意额外约束，且选民的效用属于一类非常广泛的集合函数，称为$\beta$-自界函数。当$\beta=1$时，该类包含XOS（因此也包括子模和加性）效用。我们定义了称为约束核心（restrained core）的核心稳定性新推广以处理约束，并在此概念下考虑效用的乘法逼近。我们的主要结果如下：若在约束条件下对委员会全局优化纳什福利的光滑版本，所得委员会位于$e^{\beta}$-近似约束核心内，适用于$\beta$-自界效用及任意约束。由此，即使对加性效用（因子为$e$），我们首次获得了带任意额外约束的稳定性常数逼近；即使无约束，也首次分析了XOS函数下纳什福利的稳定性。我们通过以下结果补充这一正面结论：即使对于批准效用且仅含一个额外约束，当$c<16/15$时，$c$-近似约束核心可能为空。此外，即使无约束，对$\beta$-自界函数而言，逼近中对$\beta$的指数依赖不可避免。随后，我们针对更简单的效用函数类别和约束类型给出了改进且紧的逼近结果。我们还提出了约束核心向带约束扩展合理表示（extended justified representation）的推广，并展示了拟阵约束下的存在性结果。最后，我们将结果推广至候选人数任意且无额外约束的设定。我们的技术不同于以往分析，具有独立意义。