In an approval-based committee election, the task is to select a committee of up to $k$ candidates from a set of $m$ candidates based on the preferences of $n$ voters, each of whom approves a subset of the candidates. A central open question is whether there always exists a committee in the core, a stability notion capturing proportional representation. We prove core non-emptiness for all approval-based committee elections with at most five voters. The proof is based on affine monoid methods and shows that, for $n\le5$, every fractional committee admits a deterministic rounding to an integral committee that preserves each voter's utility up to floors. We extend our argument to the weighted voter setting, which implies core existence for instances with up to five distinct approval sets. In all these cases, a core committee can be computed in polynomial time. We show that our technique cannot be extended as-is: our rounding method breaks down for $n=6$, and for $n=3$ when applied to more general models with additive valuations or non-unit candidate costs.

翻译：在基于认可投票的委员会选举中，任务是从 $m$ 名候选人集合中，根据 $n$ 名选民的偏好（每位选民认可一个候选人子集）选出至多 $k$ 名候选人组成的委员会。一个核心的开放问题是：是否总存在一个处于“核心”（一种体现比例代表性的稳定性概念）中的委员会？我们证明，在最多五位选民的所有基于认可投票的委员会选举中，核心非空。该证明基于仿射幺半群方法，并表明：对于 $n\le 5$，每个分数委员会都存在一个确定性取整方法，将其转化为整数委员会，且每位选民的效用向下取整后保持不变。我们将论证扩展至加权选民设置，这意味着在最多五个不同认可集合的实例中核心存在。在所有这些情况下，核心委员会可在多项式时间内计算得出。我们指出，我们的技术无法直接推广：取整方法在 $n=6$ 时失效，且对于具有可加估值或非单位候选人成本的更一般模型，在 $n=3$ 时亦不适用。