Core Existence in Approval-Based Committee Elections with up to Five Voter Types
This resolves a long-standing open question for the small-voter case, providing the first proof of core existence for up to five voters, but the result is incremental as it does not extend to larger voter counts.
The authors prove that in approval-based committee elections, a core committee always exists when there are at most five voters (or five distinct approval sets), and can be found in polynomial time. They show the method fails for six voters or for three voters under more general models.
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.