The Complexity of Membership, Uniqueness, and Counting for Optimal Proportional Approval Voting Committees

Proportional Approval Voting (PAV) chooses committees that maximize a sum of harmonic utilities. We study the set of maximizing committees: whether a candidate belongs to some or all of them, whether the optimum is unique, and how many optima exist. When the committee size is part of the input, the three decision problems are $Δ_2^\mathrm{P}$-complete. Uniqueness remains hard for instances with at most two optimal committees. Counting optimal committees is $\#\!\cdot\!\mathrm{OptP}$-complete under metric reductions: every function $f$ in this class reduces to an election with exactly $f(x)+1$ optimal committees. The reductions encode satisfying assignments directly as committees and use harmonic marginal rewards to realize binary objectives with polynomially many voters. Each satisfying assignment has a unique committee representation, and fixed clause ballots give these representations the same clause score. We also prove Turing equivalence with $\#\mathrm{SAT}$ and show that membership of the counting problem in $\#\mathrm{P}$ would imply $\\mathrm{NP}=\mathrm{coNP}$.