Strengthening Full Justified Representation: Efficient Verification and Computation

2026-08-11

AI Summary

This work addresses the computational complexity of verifying Fully Justified Representation (FJR) in approval-based committee elections, which is shown to be coNP-complete. To strengthen proportionality guarantees, the paper introduces a new axiom, FJR+, and proposes the Residual-Budget Greedy (RBG) algorithm—the first method capable of both satisfying and verifying FJR+ in polynomial time. By integrating RBG with the sequential Phragmén rule, the resulting voting rule consistently satisfies FJR+ and the core property; moreover, it ensures priceability whenever at least $k$ candidates receive approvals. The framework is further extended to participatory budgeting settings with arbitrary project costs, yielding an efficient algorithm that computes outcomes satisfying the cost-aware variant of the core.

Abstract

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

Problem

• Full Justified Representation
• Approval-based Committee Elections
• Proportionality Axioms
• Polynomial-time Verification
• Participatory Budgeting

Innovation

• FJR+
• polynomial-time verification
• Residual-Budget Greedy
• priceable outcome
• participatory budgeting