Strengthening Full Justified Representation: Efficient Verification and Computation

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arXiv:2608.11500v1 Announce Type: cross 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 Phragm\'en 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.

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