# Strengthening Full Justified Representation: Efficient Verification and Computation

arXiv: https://arxiv.org/abs/2608.11500 · HTML: https://mlsanity.com/paper/2608.11500/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 32 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 28 of 28)

- [Justified Representation in Approval-Based Committee Voting](https://mlsanity.com/paper/1407.8269/) (2014)
- [Proportional Participatory Budgeting with Additive Utilities](https://mlsanity.com/paper/2008.13276/) (2020)
- [Proportional Justified Representation](https://mlsanity.com/paper/1611.09928/) (2016)
- [Robust and Verifiable Proportionality Axioms for Multiwinner Voting](https://mlsanity.com/paper/2302.01989/) (2023)
- [Proportionality and the Limits of Welfarism](https://mlsanity.com/paper/1911.11747/) (2019)
- [Proportional Aggregation of Preferences for Sequential Decision Making](https://mlsanity.com/paper/2306.14858/) (2023)
- [The Core of the Participatory Budgeting Problem](https://mlsanity.com/paper/1610.03474/) (2016)
- [Temporal Fairness in Multiwinner Voting](https://mlsanity.com/paper/2312.04417/) (2023)
- [Phragm'en's Voting Methods and Justified Representation](https://mlsanity.com/paper/2102.12305/) (2021)
- [Temporal Elections: Welfare, Strategyproofness, and Proportionality](https://mlsanity.com/paper/2408.13637/) (2024)
- [Proportional Rankings](https://mlsanity.com/paper/1612.01434/) (2016)
- [Verifying Proportionality in Temporal Voting](https://mlsanity.com/paper/2502.05949/) (2025)
- [Strengthening Proportionality in Temporal Voting](https://mlsanity.com/paper/2505.22513/) (2025)
- [Not in My Backyard! Temporal Voting Over Public Chores](https://mlsanity.com/paper/2508.08810/) (2025)
- [Multiwinner Approval Rules as Apportionment Methods](https://mlsanity.com/paper/1611.08691/) (2016)
- [Proportionality in Approval-Based Participatory Budgeting](https://mlsanity.com/paper/2302.03672/) (2023)
- [Completing Priceable Committees: Utilitarian and Representation Guarantees for Proportional Multiwinner Voting](https://mlsanity.com/paper/2312.08187/) (2023)
- [Full Proportional Justified Representation](https://mlsanity.com/paper/2501.12015/) (2025)
- [The Price of Proportional Representation in Temporal Voting](https://mlsanity.com/paper/2605.11157/) (2026)
- [Fair Lotteries for Participatory Budgeting](https://mlsanity.com/paper/2404.05198/) (2024)
- [Maximum Flow is Fair: A Network Flow Approach to Committee Voting](https://mlsanity.com/paper/2406.14907/) (2024)
- [Auditing for Core Stability in Participatory Budgeting](https://mlsanity.com/paper/2209.14468/) (2022)
- [Best-of-Both-Worlds Fairness in Committee Voting](https://mlsanity.com/paper/2303.03642/) (2023)
- [An Adaptive and Verifiably Proportional Method for Participatory Budgeting](https://mlsanity.com/paper/2310.10215/) (2023)
- [Justified Representation: From Hare to Droop](https://mlsanity.com/paper/2508.00811/) (2025)
- [An Axiomatic Analysis of Proportionality Notions in Approval-Based Multiwinner Voting](https://mlsanity.com/paper/2605.04612/) (2026)
- [A Polynomial-Time Rule Satisfying Full Justified Representation](https://mlsanity.com/paper/2608.05397/) (2026)
- [Full Justified Representation under Hare and Droop Quotas in Polynomial Time](https://mlsanity.com/paper/2608.05417/) (2026)

## Built on by (most cited first, 1 of 1)

- [Proportional Representation in Temporal Voting with Ranked Preferences](https://mlsanity.com/paper/2609.30555/) (2026)

## Abstract (arXiv)

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.