A Polynomial-Time Rule Satisfying Full Justified Representation

Fabian Frank; Jannik Peters

REVIEW 3 major objections 5 minor 299 references

A Polynomial-Time Rule Satisfying Full Justified Representation

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proves that FJR-GJCR, a polynomial-time variant of the Greedy Justified Candidate Rule, always returns a committee of at most k candidates satisfying Full Justified Representation, and that MES committees cannot always be…

desk verdict Resolves the polynomial-time FJR open question with a sound main proof; the polynomial-time proof has a correctable gap and the EJR+ corollary needs work. read the letter →

arxiv 2608.05397 v1 pith:QQGSQ2VG submitted 2026-08-05 cs.GT

Fabian Frank , Jannik Peters This is my paper · ORCID

classification cs.GT

keywords approval-basedmultiwinnervotingfulljustifiedrepresentationextendedplusgreedycandidaterulemethodofequalsharesproportionalpolynomial-timecommitteeselectionvirtualpaymentscheme

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Approval-based multiwinner voting asks how to pick a committee of fixed size so that no large bloc of voters approving the same candidates is left without proportional representation. Full Justified Representation (FJR) is one of the strongest forms of that guarantee: whenever a sufficiently large group all approves at least $\beta$ candidates in some set $T$, some voter in the group must approve $\beta$ candidates on the committee. Before this paper, FJR was only known to be achievable by an algorithm that could take exponentially long. The paper resolves this by proposing FJR-GJCR, a polynomial-time variant of the Greedy Justified Candidate Rule that always returns an FJR committee of size at most $k$. Its payment scheme charges voters according to how many approved candidates they still lack, rounded to common denominators so every number stays rational and polynomial-size. The paper also shows that committees returned by the method of equal shares cannot always be extended to satisfy FJR, so a dedicated rule is genuinely needed.

What carries the argument

The carrying object is a virtual payment scheme grafted onto the Greedy Justified Candidate Rule. At each target level $\ell$, a candidate $c$ is buyable when the total over its approvers of the quotient $\frac{1-p(i)}{\ell-|A_i\cap W|}$ is at least $n/k$, where $p(i)$ is voter $i$'s already spent budget and $|A_i\cap W|$ their number of approved candidates in the current committee; buying $c$ charges voters a rounded fraction of that quotient, with all payments multiples of $\frac{1}{k(k+1)}$. The rounding does two jobs: each purchase provably spends more than $\frac{n}{k+1}$, bounding the committee size by $k$, and every intermediate number stays rational with polynomial-size representation. The FJR proof maintains a potential for every weakly $(\beta,T)$-cohesive group — a voter set $N'$ whose members each approve at least $\beta$ candidates in $T$ while $|N'|\ge |T|n/k$ — namely that while all group members still have fewer than $\beta$ approved candidates and the algorithm is still running at level $\ell\ge\beta$, the group's remaining payments toward unbought candidates in $T$ never drop below $|T\setminus W|\cdot n/k$, forcing some member to reach $\beta$ approvals.

What would settle it

Run FJR-GJCR on an instance that contains a weakly $(\beta,T)$-cohesive group and inspect the returned committee $W$: the central claim would be refuted by any output where every member of that group has fewer than $\beta$ candidates in $W$. A more local falsifier would be an intermediate step in the algorithm, while the level is still at least $\beta$, at which $\sum_{t\in T\setminus W}\sum_{i\in N'\cap N_t}\frac{1-p_i}{\beta-|A_i\cap W|}$ is strictly smaller than $|T\setminus W|\cdot n/k$, since the proof's invariant says that cannot happen.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that Full Justified Representation is compatible with efficient computation: Algorithm 1 runs in time polynomial in $n$, $m$, and $k$, selects at most $k$ candidates, and satisfies FJR for every approval profile. The rule also satisfies Extended Justified Representation Plus (EJR+), showing FJR and EJR+ are compatible. As a second contribution, Proposition 2 provides an explicit instance where the method of equal shares selects $k$ candidates and no extension of its committee can meet FJR, answering an open question negatively.

Load-bearing premise

The proof depends on a potential invariant: for every weakly $(\beta,T)$-cohesive group whose members all still have fewer than $\beta$ approved candidates while the algorithm runs at level $\ell\ge\beta$, the group's remaining virtual payments toward the unbought candidates in $T$ must stay at least $|T\setminus W|\cdot n/k$; if the invariant ever failed, the algorithm could stop at level $\beta$ with none of those voters represented.

Editorial extensions

If this is right

FJR committees can now be produced efficiently for every instance, converting an open question into a polynomial-time algorithm.
The same committee simultaneously satisfies EJR+, so FJR and EJR+ can be met without sacrificing either.
Because MES committees cannot always be extended to FJR, any guarantee at the FJR level has to come from a rule built for it, not from post-processing MES.
Replacing $n/k$ by $\frac{n}{k+1}+\frac{1}{k(k+1)}$ yields a Droop-style version of FJR with the same polynomial-time behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The quotient $\frac{1-p(i)}{\ell-|A_i\cap W|}$ is a voter-specific price per missing approved seat; an editor's reading is that any purchase rule which locally minimizes such ratios may inherit FJR, so the invariant might transfer to other greedy payment rules.
The non-extension result for MES suggests a broader negative landscape: other rules whose purchases are constrained by a fixed per-voter budget may also resist patchings, so a natural next step is to classify which priceable rules can be extended to FJR.
Since all payments are multiples of $\frac{1}{k(k+1)}$, scaling by $k(k+1)$ turns the algorithm into integer arithmetic, which could make it practical on large election data.
The Droop-FJR variant sketched at the end points toward a testable conjecture: the potential-invariant argument should carry over to participatory budgeting whenever an 'up to one project' version of FJR is used.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper settles the open question of whether a committee satisfying Full Justified Representation can be computed in polynomial time. The main theorem is essentially right: I checked the potential function argument in Theorem 1 and it holds, with one small boundary case that is easy to patch (when a voter reaches beta and the denominator would hit zero). The algorithm itself, FJR-GJCR, is a genuinely new combination of the GJCR framework with a payment-scheme rounding trick, and the FJR proof is a real contribution. The negative result about MES extensions also checks out with the numbers I ran.

The soft spots are real but not fatal. Theorem 2 contains a false statement: each p_l(c) does not have denominator k(k+1) as claimed, because the sum over voters has denominators l - |A_i∩W| that range up to k, so the common denominator can be k(k+1)*lcm(1..k). This is still polynomially many bits: lcm has O(k) bits and k <= m, so the theorem is true and the fix is routine, but the proof as written is wrong. A referee should catch this. Corollary 1 (EJR+) is underargued: the claim that adding clones does not change the execution of FJR-GJCR is asserted without proof, and it is not obvious. The MES example in Proposition 2 says MES selects k candidates but the construction yields 76 out of k=77; the argument still works, but the text should be corrected. The informal Proposition 1 is fine as motivation.

Who is this for? Researchers in approval-based multiwinner voting and computational social choice. The central result is significant and the proof technique is inventive. The issues are patchable, not fundamental. I would send it to peer review and ask for a corrected Theorem 2 proof, a proper proof of the clone-independence claim or a reference, and a cleanup of the MES example. After that, it would be a strong paper.

Referee Report

3 major / 5 minor

Summary. The paper proposes a new approval-based multiwinner voting rule, FJR-GJCR (Algorithm 1), which is designed to satisfy Full Justified Representation (FJR). The central claims are that the rule selects at most k candidates (Lemma 1), satisfies FJR (Theorem 1), is polynomial-time computable (Theorem 2), and also satisfies EJR+ (Corollary 1). The paper additionally presents an instance showing that a committee returned by the Method of Equal Shares cannot always be extended to an FJR committee (Proposition 2).

Significance. If the results are correct, the paper resolves a central open question in the area by giving the first polynomial-time rule satisfying FJR. The proof of Theorem 1 is, in substance, a sound potential-function argument, and the MES counterexample in Proposition 2 is convincing and correctly shows that the natural extension question has a negative answer. The paper is also careful in that the main FJR guarantee is derived directly from the algorithm's payment schedule rather than from fitted parameters or circular assumptions. The main weakness is that the proof of Theorem 2, which is half of the paper's central claim, contains a false statement about the denominators of the quantities p_ell(c).

major comments (3)

[Section 3, Theorem 2] The proof of Theorem 2 claims that each p_ell(c) is a rational number with denominator k(k+1). This is false. Since p(i) is a multiple of 1/(k(k+1)), each summand (1-p(i))/(ell - |A_i cap W|) has denominator k(k+1) * d_i for some d_i in {1,...,k}, so p_ell(c) has a denominator dividing k(k+1) * lcm(1,...,k). For example, with k=4, ell=4, p(i)=0 for two voters with |A_i cap W| equal to 0 and 1, the sum is 1/4 + 1/3 = 7/12, and 12 does not divide 20 = k(k+1). Therefore the polynomial-time claim is not established as written. A corrected proof using the common denominator L = k(k+1) * lcm(1,...,k) is plausible, because L has O(k) bits and k <= m, but the argument in the manuscript must be rewritten.
[Section 3, Theorem 1 proof] The inductive step in Theorem 1 divides by beta - |A_i cap W_j| - 1. If a voter in the weakly cohesive group has beta - |A_i cap W_j| = 1 before a candidate is bought, this denominator is zero after the purchase. The proof does not address this boundary case. The gap is repairable by arguing under a contradiction assumption that no voter in the group has yet reached beta approvals, or by explicitly removing voters who become satisfied from the potential, but the current text is formally incomplete.
[Corollary 1] The proof of Corollary 1 asserts that adding clones of an unelected candidate does not change the execution of FJR-GJCR. This independence-of-cloning-losers property is not proved, and it is not obvious from the pseudocode, since Algorithm 1 does not specify a tie-breaking rule for choosing among candidates with equal p_ell(c). If the rule is intended to select a candidate with maximal p_ell(c), the clone-invariance claim needs a proof or a carefully chosen tie-breaking convention; without this, the EJR+ conclusion is not justified.

minor comments (5)

[Definition 2] The quantifier "for every ell in [k]" in Definition 2 is unused in the condition and conclusion; the definition only depends on beta and the weakly cohesive group. This should be cleaned up for clarity.
[Algorithm 1] The pseudocode in lines 8-10 does not specify which candidate c is selected when multiple candidates satisfy p_ell(c) >= n/k, although the prose above the algorithm says the candidate with largest total payment is added. The pseudocode and prose should be made consistent.
[Section 3, Proposition 2] The introduction to Proposition 2 says the instance is one "in which the method of equal shares selects k candidates," but the proof shows that MES selects 76 candidates while k=77. The wording should be corrected to say that MES selects a committee of size at most k or that it leaves one seat unfilled.
[Proposition 2 proof] In the last paragraph of the proof, "approves 27 voters in the extension" should read "approves 27 candidates in the extension."
[Throughout] There are several typographical errors, including "could bought" in the first sentence of Theorem 1, "reasearch" in the acknowledgments, and "intersting" and "approp-priatly" in the conclusion. These should be corrected in a revision.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the FJR proof is self-contained; only minor auxiliary self-citations appear.

full rationale

The central claim that Algorithm 1 satisfies FJR is proved in Theorem 1 through an explicit potential-function invariant over the algorithm's own payment schedule. The base case and inductive step are derived directly from the definition of weakly cohesive groups and the payment update rule; no fitted parameter is used, and the FJR conclusion is not assumed at any point. Lemma 1 bounds the committee size by the same payment accounting. Proposition 2, the MES non-extension result, is a standalone construction with no reliance on the paper's own prior work. The only self-citations occur in auxiliary or motivational places: the informal Proposition 1 is explicitly labeled 'informally' and is not a proof step, and Corollary 1 cites Peters [2024] for the EJR+ argument but also reproduces the clone-based reasoning and reduces it to the independently proven Theorem 1. These citations are not load-bearing for the paper's principal result. The denominator issue in Theorem 2's proof (p_ell(c) may require denominator k(k+1)*lcm(1..k) rather than k(k+1)) is a correctness gap in the polynomial-time argument, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters and no invented entities are introduced. The central proof relies on standard mathematical tools (pigeonhole principle) and on the paper's own definitions. The informal Proposition 1 cites prior work as motivation only, not as a load-bearing step. The clone-invariance property asserted in Corollary 1 is an additional unproved assumption.

assumptions (5)

domain assumption Standard approval-based multiwinner voting model: voters have approval sets A_i, committee size k, and each candidate in MES has price n/k.
Used throughout, including the definitions of FJR, EJR+, and the MES counterexample.
standard math Pigeonhole principle.
Used in Theorem 1 to turn a lower bound on the sum of potentials over T\W_j into the existence of at least one candidate t with p_beta(t) >= n/k.
domain assumption Peters et al. [2021, Lemma 2] (informal Proposition 1): a weakly cohesive group can afford to buy T at cost-per-utility at most 1/beta.
Used only as intuition for the payment scheme; the formal proof of Theorem 1 is self-contained and does not invoke this lemma.
ad hoc to paper Independence of cloning losers for FJR-GJCR (adding clones of an unelected candidate does not change the execution).
Asserted without proof in the proof of Corollary 1; needed to derive EJR+ from FJR.
domain assumption The Method of Equal Shares behaves as computed in the constructed instance of Proposition 2.
The counterexample relies on the standard MES definition and on arithmetic for budget depletion; the paper provides the calculations but no machine verification.

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Pith. "Pith review of A Polynomial-Time Rule Satisfying Full Justified Representation." pith.science (2026). https://pith.science/paper/QQGSQ2VG

@misc{pith2026260805397,
  author       = {Pith},
  title        = {Pith review of: A Polynomial-Time Rule Satisfying Full Justified Representation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQGSQ2VG}},
  note         = {Machine review of arXiv:2608.05397}
}

abstract

In approval-based multiwinner voting, voters submit an approval ballot on which basis a committee of fixed size $k$ has to be selected. Compared to single-winner voting this makes it possible to represent minorities in the selection of the committee as well. Thus, one of the most desirable goals of multiwinner voting is to satisfy proportional representation. In this paper we propose a variant of the Greedy Justified Candidate rule that satisfy the notion of Full Justified Representation (FJR). This variant can be computed in polynomial time answering the open question of whether an FJR committee can always be computed efficiently. Additionally, we answer an open question whether committees returned by MES can always be extended in a way to satisfy FJR negatively.

Figures

Figures reproduced from arXiv: 2608.05397 by the authors.

**Figure 1.** A visualization of Example 1 showing that MES fails FJR. Where MES seemingly went wrong is when it bought one of the c5, c6, c7 candidates. While each of them would give a locally optimal cost-per-utility ratio of 1 3 , this does not take the global cost-per-utility ratio into consideration. In fact, the first 15 voters at this moment had a strictly smaller global cost-per-utility ratio of 2 15 . Our idea is now, th… view at source ↗

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Haris Aziz and X. Lu and M. Suzuki and Jeremy Vollen and Toby Walsh , booktitle = proc #. Fair Lotteries for Participatory Budgeting , year =
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Jannik Peters , booktitle = proc #. Online Elicitation of Necessarily Optimal Matchings , year =
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Putting a Compass on the Map of Elections , year =

Niclas Boehmer and Robert Bredereck and Piotr Faliszewski and Rolf Niedermeier and Stanis. Putting a Compass on the Map of Elections , year =
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M. Jaworski and Piotr Skowron , booktitle = proc #. Evaluating committees for representative democracies: the distortion and beyond , year =
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Guide to Numerical Experiments on Elections in Computational Social Choice , year =

Niclas Boehmer and Piotr Faliszewski and. Guide to Numerical Experiments on Elections in Computational Social Choice , year =. doi:10.24963/ijcai.2024/881 , pages =

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Markus Brill and E. Markakis and G. Papasotiropoulos and Jannik Peters , booktitle = proc #. Proportionality Guarantees in Elections with Interdependent Issues , year =
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M. D. Margin of Victory for Weighted Tournament Solutions , year =
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Lu and Jannik Peters and Haris Aziz and X

X. Lu and Jannik Peters and Haris Aziz and X. Bei and Warut Suksompong , date-added =. Approval-based voting with mixed goods , volume =. Social Choice and Welfare , keywords =. 2024 , bdsk-file-1 =

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Micha , date-added =

P. Micha , date-added =. Fair and Efficient Social Decision Making , year =
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Approval-Based Committee Voting in Practice: A Case Study of (Over-)Representation in the

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Completing Priceable Committees: Utilitarian and Representation Guarantees for Proportional Multiwinner Voting , year =

Markus Brill and Jannik Peters , booktitle = proc #. Completing Priceable Committees: Utilitarian and Representation Guarantees for Proportional Multiwinner Voting , year =. doi:10.1609/aaai.v38i9.28808 , keywords =

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The Maximin Support Method: An Extension of the

Luis S. The Maximin Support Method: An Extension of the. Mathematical Programming , number =
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Approval-Based Apportionment , volume =

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Janson and Martin Lackner, , date-added =

Markus Brill and Rupert Freeman and S. Janson and Martin Lackner, , date-added =. Phragm\'. Mathematical Programming , number =. 2024 , bdsk-file-1 =

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A Quantitative and Qualitative Analysis of the Robustness of (Real-World) Election Winners , year =

Niclas Boehmer and Robert Bredereck and Piotr Faliszewski and Rolf Niedermeier , booktitle =. A Quantitative and Qualitative Analysis of the Robustness of (Real-World) Election Winners , year =

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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