Core Existence in Approval-Based Committee Elections with up to Five Voter Types

Abstract: 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.

• The paper proves that every approval-based committee election with at most five voters or distinct approval sets has a core committee, including a Pareto-optimal one, and extends the result to weighted voters.
• It establishes normality of an affine committee-election monoid for up to five voter types, enabling fractional core outcomes to be rounded into integral committees without reducing any voter’s utility below its floor.
• It gives a polynomial-time construction using approximate Lindahl equilibria, determinant bound L₅ = 60, and LP-based rounding, while showing the method fails at six voters and identifying open cases for core existence.

The core stability question in approval-based committee elections

In an approval-based committee election, $n$ voters each approve a subset of $m$ candidates, and a committee of size at most $k$ must be selected. A committee is in the core if no coalition $S \subseteq N$ can use its proportional budget share $\frac{|S|}{n}k$ to select a subcommittee that every member strictly prefers. Whether the core is always non-empty has been open since the concept was introduced for this setting by Aziz et al. [(Minkin et al., 2016)-style work cited as ABC+16a]. Known partial results include existence for $k \le 8$ (via PAV), for $m \le 15$, and constant-factor approximations of the core; PAV itself fails the core already at $n = 3$.

This paper proves that the core is non-empty for every instance with at most five voters, and more generally with at most five distinct approval sets ("voter types"). Moreover, a core committee — and even a Pareto-optimal one — can be computed in polynomial time. The result also holds under arbitrary voter weights, which is what enables the extension from few voters to few voter types by merging same-type voters into weighted meta-voters.

From the fractional core to the integral core

The fractional core (where coalitions deviate using fractional committees) is always non-empty, via Lindahl equilibria; the paper supplies a self-contained derivation by perturbing utilities to be strictly monotone and taking limits of Foley's fixed-point theorem, since zero valuations and supply caps are not directly covered by Foley's original assumptions.

The key bridge is the notion of integralizability: an instance $(c,k)$ is integralizable if every integer utility vector achievable by a fractional committee is achievable by an integral one. On an integralizable instance, rounding down the utilities of a fractional core committee yields an integral core committee: any blocking coalition would need to improve each member by at least $1$, which exceeds the fractional core outcome's utility. The paper also gives elementary instances of integralizability ($m$0, and candidate-interval domains via total unimodularity).

Normality of the committee election monoid

Candidates are grouped into types $m$1 (the set of approving voters). The paper defines an affine monoid $m$2 in $m$3 generated by vectors encoding "select one candidate of type $m$4", "unused supply", "unused seats", and per-voter utility slack. Membership in $m$5 exactly characterizes pairs $m$6 where $m$7 is integral-committee-feasible, while $m$8 corresponds to fractional feasibility. Hence:

Theorem (main technical result). For $m$9, $k$0 is integrally closed (normal): every fractionally feasible integer utility vector is integrally feasible.

Normality was first confirmed computationally with the Normaliz program (68 generators for $k$1, verified in under 0.05 s), but the authors give an explicit human-readable proof via reduction lemmas: any counterexample can be reduced by subtracting generators until it is minimal, and minimal counterexamples satisfy strong structural constraints — unit supplies forming an antichain, all witnesses strictly fractional, $k$2, $k$3, hence $k$4. For $k$5 no counterexample exists vacuously; $k$6 reduces to a graph-theoretic argument on 4-cycles; $k$7 requires a case analysis showing $k$8 is impossible and deriving a counting contradiction for $k$9. A corollary is a clean rounding statement: for $S \subseteq N$0, every fractional committee rounds to an integral committee preserving each voter's floored utility while changing each coordinate by less than $S \subseteq N$1.

The main theorem has been formally verified in Lean 4 within the ABCVotingLean project, independently of Normaliz.

Polynomial-time computation

Computing an exact fractional core point efficiently remains open, so the algorithm uses the approximate Lindahl-based convex program of Kluiving–Peters, adapted so that the only relaxation is a budget overshoot of $S \subseteq N$2. A denominator argument shows that the minimum fractional committee size realizing an integer utility vector lies in $S \subseteq N$3, where $S \subseteq N$4 is the lcm of determinants of binary matrices of order at most $S \subseteq N$5; crucially $S \subseteq N$6. Choosing $S \subseteq N$7 ensures the rounded-down utility vector is feasible within budget exactly $S \subseteq N$8. The vector is then implemented integrally via a greedy LP-based rounding procedure, and core membership follows because any blocking coalition would need to exceed the approximate fractional solution's utilities. Pareto optimality is added either by solving a small ILP (at most 31 variables, solvable in polynomial time by Lenstra's theorem) maximizing utilitarian welfare subject to the achieved utility floor, or by a serial-dictatorship-style greedy increase of utilities checked through LPs.

Limits of the technique

The approach does not extend as-is beyond its domain:

• Six voters. The monoid is not normal for $S \subseteq N$9: two explicit non-integralizable instances exist (two disjoint triangles with $\frac{|S|}{n}k$0, and four candidates covering pairs of triples with $\frac{|S|}{n}k$1). Notably, both instances still have non-empty cores, so non-normality does not imply core emptiness.
• Droop quota. With budgets just below $\frac{|S|}{n}k$2, the rounding property fails already at $\frac{|S|}{n}k$3, $\frac{|S|}{n}k$4.
• Additive valuations. A single Condorcet cycle with $\frac{|S|}{n}k$5 breaks normality at $\frac{|S|}{n}k$6.
• Non-unit costs / participatory budgeting. Three candidates of cost 2 approved pairwise break normality at $\frac{|S|}{n}k$7.

The appendix additionally shows MES violates the core for $\frac{|S|}{n}k$8 under some tie-breaking, gives a dedicated MES-like tie-breaking rule that always returns a Pareto-optimal core committee for three voters, and exhibits a nine-voter instance on which MES fails the core regardless of tie-breaking.

Limitations and open questions

The existence guarantee is confined to at most five voter types; nothing in the paper extends it to six, and the monoid method provably breaks there. The computational results depend on the approximation scheme for Lindahl equilibria and on the determinant bound $\frac{|S|}{n}k$9, both specific to the small-$k \le 8$0 regime. The paper leaves several questions open: whether all $k \le 8$1 violations of normality correspond to instances that nonetheless admit core committees (a potential route to extending existence guarantees); whether the affine-monoid rounding technique applies to other restricted preference domains beyond candidate-interval instances; and whether it can establish existence of strengthenings such as stable priceability or Lindahl priceability, whose connection to utility vectors is less direct.

Conclusion

By proving normality of the committee election monoid for up to five voter types, the paper converts the guaranteed existence of fractional core committees into exact integral core committees, computable in polynomial time and enforceable to be Pareto optimal. Combined with prior bounds, any counterexample to core existence in approval-based committee elections must involve at least six voter types, sixteen candidates, and committee size at least nine — sharply delimiting where the long-standing open problem can still have a negative answer.