Ai and Tao Prove Fair Election Committee Selection Is Harder Than Expected

A new paper by Ai and Tao, published on arXiv, tackles the computational complexity of fairness in multiwinner elections. In approval-based voting, voters approve a subset of candidates, and a committee of size k is selected. Two key axioms—strong justified representation (SJR) and average justified representation (AJR)—ensure that groups of voters with shared preferences get proportional representation. SJR requires that every voter in an ℓ-cohesive group have satisfaction at least ℓ, while AJR only demands the average satisfaction meet that threshold. The authors prove that deciding whether any committee satisfies SJR is Θ₂ᵖ-complete, placing it at the second level of the polynomial hierarchy but only using non-adaptive NP queries. In contrast, AJR is Σ₂ᵖ-complete, meaning it is strictly harder—requiring adaptive NP queries and lying higher in the hierarchy.

Beyond the complexity results, the paper contributes two theoretical byproducts. First, it shows that adding one more adaptive query to an NP oracle on top of polynomially many non-adaptive NP queries does not increase computational power—the resulting class remains Θ₂ᵖ. Second, it constructs a novel set system that simplifies reductions from typical satisfiability problems like 3SAT. For practitioners, the findings have clear implications: SJR-based committee selection is more tractable and could be implemented via SAT-based tools, while AJR, though a weaker fairness notion, is fundamentally harder to verify. This guides the design of computationally feasible democratic processes in settings like participatory budgeting, corporate board elections, or AI-based recommendation systems that aim for fair group outcomes.