Fully Justified Representation (FJR)

• Fully Justified Representation is a fairness axiom that guarantees proportional representation for weakly cohesive groups in collective decision-making.
• It generalizes previous notions like JR, PJR, and EJR by relaxing unanimity requirements and ensuring individual representation across diverse domains.
• Algorithmic frameworks such as the Greedy Cohesive Rule implement FJR, though challenges remain in computational complexity and efficient approximation methods.

Fully Justified Representation (FJR) is a rigorous proportional fairness axiom operating across a variety of collective decision-making domains, including clustering, participatory budgeting, multiwinner voting, and temporal voting. FJR generalizes previous proportional representation notions—such as Justified Representation (JR), Proportional JR (PJR), and Extended JR (EJR)—by demanding guarantees for broader classes of cohesive groups, strengthening individual representation and relaxing the unanimity requirements imposed by earlier axioms. FJR is motivated by democratic fairness ideals and underpins algorithmic frameworks in discrete optimization, social choice, and resource allocation.

1. Formal Definition and Axiomatic Positioning

FJR is instantiated differently across domains but always centers on the concept of "weakly cohesive" groups—coalitions large enough to deserve proportional representation, where group members share partial but not necessarily unanimous preferences or attributes. In approval-based committee voting, FJR holds for any weakly $(\beta,T)$-cohesive group $N'$ (i.e., $|N'|\ge|T|\cdot\frac{n}{k}$ and $|A_i\cap T|\ge\beta$ for all $i\in N'$), requiring that at least one member obtains $\beta$ representatives approved from $T$ in the final outcome (Aziz et al., 2023). In participatory budgeting, FJR demands that any group $S$ which is large enough (in budget terms) and values a bundle $T$ sufficiently (each $u_i(T)\ge\alpha$) sees at least one member obtain utility at least $\alpha$ in the funded set (Peters et al., 2020, Aziz et al., 2024). In clustering, FJR requires that no large and cohesive group $S$ can form its own cluster with strictly lower loss for every member than the minimum among them in the candidate clustering (Caragiannis et al., 2024, Cookson et al., 1 Jan 2026). In temporal voting, FJR generalizes satisfaction bounds by allowing groups to exploit partial agreement over multiple rounds and optimizing guarantees over flexible allocations of “rounds” (Phillips et al., 28 May 2025).

FJR strictly strengthens EJR and PJR by relaxing strong unanimity (intersection) requirements and imposing more demanding individual guarantees. Formally, the hierarchy is:

$$\text{JR} \subseteq \text{PJR} \subseteq \text{EJR} \subseteq \text{FJR} \subseteq \text{Core}$$

Each inclusion is strict in general; every FJR outcome satisfies EJR, but the converse does not hold (Aziz et al., 2023, Casey et al., 1 Aug 2025, Peters et al., 2020, Aziz et al., 2024).

2. FJR Across Domains: Models and Formalisms

The table below summarizes canonical FJR definitions across prominent domains.

Domain	Cohesion Criterion	FJR Guarantee
Approval committee voting	$|N'|\ge|T|\cdot n/k$, $|A_i\cap T|\ge \beta$	$\exists i\in N': |A_i\cap W|\ge\beta$ (Aziz et al., 2023, Casey et al., 1 Aug 2025)
Participatory budgeting	$|S|/n > \text{cost}(T)/b$, $u_i(T)\ge\alpha$	$\exists i^*\in S: u_{i^*}(W)\ge\alpha$ (Peters et al., 2020, Aziz et al., 2024)
Clustering (non-centroid)	$|S|\ge n/k$	$\alpha \cdot \ell_i(S) < \min_{j\in S} \ell_j$ (Caragiannis et al., 2024)
Semi-centroid clustering	$|S|\ge n/k$, $y\in M$	$\alpha \cdot \ell_i(S,y) < \min_{j\in S} \ell_j(X)$ (Cookson et al., 1 Jan 2026)
Temporal voting	Size–round tradeoff, optimized over possible $T$	$\exists i\in S: sat_i(o)\ge\max_T \mu_S(T)$ (Phillips et al., 28 May 2025)

In multiwinner voting, Droop-FJR modifies the classic Hare quota $n/k$ to Droop’s stricter $n/(k+1)$, significantly intensifying the cohesiveness threshold (Casey et al., 1 Aug 2025).

3. Algorithmic Frameworks and Computational Complexity

Existence of FJR outcomes is established for all domains under consideration. The principle method is the Greedy Cohesive Rule (GCR), which iteratively satisfies the most deserving weakly cohesive group by removing satisfied agents and adding their requested representatives, projects, or clusters. In clustering, iterative cohesive clustering yields exact FJR in polynomial time for certain loss functions and constant-factor approximations otherwise (Caragiannis et al., 2024, Cookson et al., 1 Jan 2026). In committee voting and participatory budgeting, GCR ensures FJR but requires brute-force search, yielding exponential time in the worst case (Peters et al., 2020, Aziz et al., 2023), whereas EJR is attainable in polynomial time by budget-distribution mechanisms such as the Method of Equal Shares (Peters et al., 2020). For the Droop-FJR axiom, a corresponding GreedyCohesiveRule with Droop quota ensures existence (Casey et al., 1 Aug 2025).

Some efficient, domain-specialized approximation methods are known. GreedyCapture in clustering yields a constant-factor FJR guarantee using metric ball-growing (Caragiannis et al., 2024). In committee voting, polynomial-time best-of-both-worlds algorithms achieve EJR plus strong ex-ante fairness (Group Fair Share GFS or Strong UFS), but full ex-post FJR mechanisms with ex-ante GFS require exponentially large support (Aziz et al., 2023).

4. Lower Bounds, Impossibility, and Hierarchy Separations

FJR admits strict separation results both from weaker axioms and from the “core” which demands no group can strictly improve for all its members. In clustering, incompatibility arises when attempting to simultaneously guarantee centroid and non-centroid FJR under distinct metrics; no clustering can satisfy FJR for both metrics in strong approximation factors (Cookson et al., 1 Jan 2026). Balanced clustering (exact equal-sized clusters) poses inherent lower bounds: with weighted single-metric loss, finite approximation for FJR below $1/\sqrt{\lambda}$ is impossible (Cookson et al., 1 Jan 2026). Committee voting under Droop quotas is provably more demanding than Hare-FJR, as randomized committees and standard rules are empirically much less likely to pass the Droop-FJR test for large $n$, small $k$ (Casey et al., 1 Aug 2025). In PB, Equal Shares and Proportional Approval Voting can fail FJR even when EJR is satisfied (Peters et al., 2020). In temporal voting, FJR admits strictly more demanding satisfaction bounds than EJR by allowing flexible allocations of rounds and maximizing over min–max group guarantees (Phillips et al., 28 May 2025).

5. Practical Algorithms, Approximations, and Auditing

While exact FJR computation often requires exponential time, several scalable heuristics and constant-factor approximation algorithms exist. In clustering, GreedyCapture achieves 2-FJR for max-loss and 4-FJR for avg-loss in $O(kn)$ time by growing smallest-radius metric balls (Caragiannis et al., 2024). In semi-centroid clustering, ball-growing, greedy capture, and semi-ball-growing achieve improved approximation ratios depending on the metric combination and $\lambda$ parameter (Cookson et al., 1 Jan 2026). Efficient auditing algorithms can estimate the FJR violation of any fixed clustering, bounding the true violation within a known factor (Caragiannis et al., 2024). Participatory budgeting with binary utilities admits polynomial-time FJR through GCR and best-of-both-worlds randomization schemes yielding FJR plus budget balance up to one project (BB1) and strong UFS (Aziz et al., 2024). In fractional/lottery mechanisms, dependent rounding ensures that every sampled outcome in the lottery upholds FJR (Aziz et al., 2023).

6. Illustrative Examples and Empirical Results

Numerical and experimental analysis demonstrates the actionable impact of FJR. In clustering tasks (UCI datasets), GreedyCapture yields dramatically reduced FJR and core violations compared with $k$-means++ or $k$-medoids, with only modest accuracy loss (Caragiannis et al., 2024). In committee voting, examples delineate instances where EJR holds but FJR fails, confirming strict separation (Aziz et al., 2023, Peters et al., 2020). Participatory budgeting examples show how GCR can satisfy FJR where Equal Shares does not (Peters et al., 2020). In temporal voting, FJR-compliant outcomes can be strictly more demanding—entries that exploit cohesiveness across rounds may fail EJR but pass FJR, or conversely, expose violations that EJR does not detect (Phillips et al., 28 May 2025).

7. Open Problems and Future Directions

Efficient polynomial-time algorithms for exact FJR remain elusive across multiple domains, particularly for general loss functions in clustering and with complex (non-additive or complementary) utilities in PB (Peters et al., 2020, Aziz et al., 2023, Phillips et al., 28 May 2025, Cookson et al., 1 Jan 2026). Complexity upper bounds are not fully settled; NP-hardness or coNP-membership for verifying FJR is open (Peters et al., 2020, Aziz et al., 2024, Phillips et al., 28 May 2025). Fundamental questions surround the existence of “natural” market-inspired FJR rules that could match the simplicity and interpretability of Equal Shares or related mechanisms (Peters et al., 2020). Compatibility of FJR with ex-ante fairness notions (GFS/UFS) admits incompatibility barriers unless exponential lottery supports are permitted (Aziz et al., 2023, Aziz et al., 2024). Unification of FJR across domains—particularly in voting, budgeting, clustering, and temporal sequencing—offers avenues for cross-disciplinary methodology and theory advancement.

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FJR codifies the strongest proportional representation guarantee known to be universally achievable in diverse social choice and discrete optimization environments. It subsumes classical axioms by relaxing unanimity and amplifying individual representation, yet poses significant computational and conceptual challenges. Research continues to advance algorithmic techniques, approximation methods, and practical frameworks for realizing FJR in scalable settings.