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：暂无翻译