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Verifying Proportionality in Temporal Voting

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In temporal approval elections, checking whether an outcome gives voter groups their proportional share is coNP-complete for JR, PJR, and EJR even with only two candidates, making temporal verification strictly harder than multiwinner…

desk verdict The core coNP-hardness result for temporal JR/PJR/EJR verification is solid and the paper deserves serious refereeing, but two appendix proofs (Prop 7.4 and Thm 5.4) need real fixes before acceptance. read the letter →

arxiv 2502.05949 v1 pith:6MDEXL5D submitted 2025-02-09 cs.GT cs.AI

Edith Elkind , Svetlana Obraztsova , Jannik Peters , Nicholas Teh This is my paper · ORCID
classification cs.GTcs.AI MSC 68Q1768Q2591B12
keywords temporalvotingperpetualjustifiedrepresentationcoNP-completenessparameterizedcomplexityapprovalproportionalmonotonicpreferences
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

This paper studies temporal (perpetual) voting, where in each of a fixed number of rounds voters approve candidates and one candidate is selected, and asks how hard it is to verify that a proposed sequence of winners fairly represents groups of voters. The authors prove that verifying any of the standard proportionality axioms—JR, PJR, and EJR, as well as their weak variants—is coNP-complete, and for the strong versions this remains true even when there are only two candidates. Because verifying JR is easy in ordinary multiwinner elections, the temporal setting is strictly harder, not merely a translation of known results. The paper then maps where the hardness breaks: verification is fixed-parameter tractable in the number of voters, tractable when preferences are monotone over time, and W[1]-hard in the number of rounds for the weak axioms. A corollary is a fixed-parameter algorithm for finding EJR outcomes under extra welfare constraints and an impossibility result ruling out EJR in semi-online settings.

What carries the argument

The load-bearing object is the demand function α(N') = ⌊β(N')·|N'|/n⌋, where β(N') counts the rounds in which all members of group N' approve a common candidate; it converts group agreement into a number of rounds the group can claim, and each axiom is a check that some or all group members' satisfaction reaches or covers α(N'). The hardness proofs work by constructing elections in which a witnessing group exists exactly when a combinatorial object (clique, independent set, biclique) exists, so that verifying fairness becomes solving an NP-hard search problem. The tractability proofs exploit structural simplifications: for monotone preferences the only groups that matter are the sets N_{p,t}^z of voters sorted by satisfaction, and for two-candidate weak JR it suffices to inspect the set of 'grumpy' voters who approve only the unchosen candidate in every round.

What would settle it

A concrete test is to run the Theorem 3.2 reduction on a Maximum Edge Biclique instance with κ > |L|+|R|: the all-q outcome is claimed to violate JR exactly when the graph has a κ-edge biclique, so any algorithm that verifies JR on this two-candidate family in polynomial time would refute the theorem, as would a polynomial-time reduction from temporal JR verification to multiwinner JR verification, which would collapse the claimed separation.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that proportionality verification in temporal voting is fundamentally harder than in static multiwinner elections. For each of the six axioms (JR, PJR, EJR and their weak forms), telling whether a given outcome satisfies the axiom is coNP-complete; for JR/PJR/EJR hardness persists at |P|=2, and the proof goes through a reduction from Maximum Edge Biclique that encodes a cohesive voter group as a biclique whose edge count exceeds the demand threshold. The only polynomial cases are small candidate sets under special conditions—two candidates with non-empty approvals for JR, and monotonic preferences—while the general hardness extends to W[1]-hardness in the number of rounds for weak axioms. The same machinery yields an FPT algorithm parameterized by the number of voters for finding EJR outcomes with side constraints, a two-stage greedy cohesive rule that always returns an EJR outcome, and a proof that no semi-online rule can guarantee EJR.

Load-bearing premise

Everything in the paper depends on defining a group's demand as the floor of (rounds of full agreement times group size divided by total voters); if proportional representation were formalized with the alternative demand min{β(N'), ℓ·|N'|/n}, even JR becomes unsatisfiable and the whole complexity question evaporates.

Editorial extensions

If this is right

  • Temporal proportionality verification cannot reuse multiwinner algorithms: even JR, which is polynomial-time solvable in static elections, becomes coNP-complete here.
  • With two candidates, strong JR/PJR/EJR verification stays coNP-complete, so the hardness is not an artifact of many candidates; only weak JR with two candidates is polynomial-time solvable.
  • Monotone elections, where candidates appear over time but never leave and voter preferences are fixed, admit polynomial-time verification of all six axioms.
  • Finding an EJR outcome that also guarantees each voter a prescribed satisfaction level and maximizes utilitarian welfare is fixed-parameter tractable in the number of voters via an integer linear program.
  • No rule can satisfy EJR in the semi-online setting, where preferences arrive round by round and each winner must be chosen immediately.

Reading between the lines

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

  • Because the demand formula α(N') is inherited from earlier work and the alternative min{β(N'), ℓ|N'|/n} makes even JR unsatisfiable, the entire complexity classification is hostage to this modeling choice; a different formalization of temporal proportionality could push the hard/easy boundary in unexpected ways.
  • The reductions' insensitivity to candidate count but sensitivity to whether approval sets may be empty suggests that the practically important frontier is domain restrictions rather than |P|; the authors explicitly conjecture that hardness persists for |P|=3 with non-empty approvals.
  • The two-stage greedy cohesive rule suggests a design principle for temporal fairness: partition voters into disjoint cohesive groups and allocate rounds group by group, a decomposition that could seed approximation or fixed-parameter algorithms for welfare-constrained fair outcomes.
  • The semi-online impossibility implies that practical sequential decision systems—participatory budgeting, repeated recommendations, annual charity selection—should view full EJR as an offline-only guarantee and plan for weaker axioms when decisions cannot be deferred.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the computational complexity of verifying whether a given outcome of a temporal (perpetual) approval election satisfies the temporal analogues of justified representation (JR), proportional justified representation (PJR), and extended justified representation (EJR), both in their weak and strong forms. The main results are coNP-completeness of verification for all three strong axioms even with two candidates (Theorem 3.2), coNP-completeness for the weak axioms (Theorem 3.1), FPT/XP results with respect to the number of voters and the combined parameter (m, ℓ), polynomial-time verification for monotonic preferences (Theorems 6.1 and 6.2), a two-stage greedy rule that provably returns EJR outcomes (Theorem 7.1), an ILP formulation for EJR outcomes with additional voter-specific lower bounds (Theorem 7.2, Corollary 7.3), and an impossibility claim for EJR in the semi-online setting (Proposition 7.4).

Significance. The central verification-hardness results are carefully argued and constitute a genuine conceptual contribution: they show that temporal verification is strictly harder than multiwinner verification, since JR verification is polynomial-time in the multiwinner model but coNP-complete here even with two candidates. The paper also contains several useful positive results, including FPT in the number of voters, polynomial algorithms for monotonic preferences, and a constructive EJR rule. The reductions are self-contained and use standard source problems, and the paper is honest about its modeling assumptions, explicitly discussing the alternative demand function and why it is unsatisfiable. However, three secondary technical contributions have proof gaps: Proposition 7.4, the W[1]-hardness proof of Theorem 5.4, and the ILP formulation in Theorem 7.2. These gaps are real but appear fixable, so the appropriate decision is a major revision rather than rejection.

major comments (3)
  1. [§7.3, Appendix D.2] The proof of Proposition 7.4 is invalid. The step "By symmetry, we can assume without loss of generality that in the first k rounds, we select o_t = p_t for t∈[k]" is not justified for an arbitrary semi-online rule, which need not be neutral and may break the symmetry of the constructed instance. More seriously, the fixed instance does not have the property that every w-JR outcome fails EJR: for k=4, the outcome o=(p1,p5,p6,p7,p2,p3,p4,p8) satisfies w-JR, and voter 1 in the group [4] has satisfaction 2 = α([4]), so EJR is not violated for the critical group. The impossibility claim requires an adversary argument that reveals preferences round-by-round depending on the rule's past choices; the current proof only analyzes one particular w-JR outcome and cannot be repaired by the stated symmetry argument.
  2. [§5.4 (Theorem 5.4), Appendix B.2] The reduction for the case n′/k > k is not correct. The n′′ added voters are described as approving "every candidate", which includes the dummy candidate in each round. Since the outcome o consists entirely of dummy candidates, these added voters have satisfaction ℓ > 0 and hence cannot be part of any group witnessing a w-JR violation, which requires every member of the group to have satisfaction 0. Thus the group consisting of the k clique voters together with the n′′ added voters is not a valid witness, and the "clique ⇒ violation" direction fails; the clique alone is too small to reach the demand threshold n/k. The converse direction also lacks a proof that any witness must contain at least k vertex voters. The proof should be repaired, e.g., by making the added voters approve all non-dummy candidates and proving n′′ < n/k so that the all-added-voters group is too small; as written, Theorem 5.4 is unsupported.
  3. [§7.2, Appendix D.1] The ILP in the proof of Theorem 7.2 does not enforce that a selected candidate is actually present in the round type. For each round type τ, variables x_{p,τ} exist for every p∈P′, but a candidate p_V that is not approved by anyone in rounds of type τ (i.e., no original candidate has profile V in that round) can still be assigned x_{p,τ}=κ_τ, satisfying constraint (2) without contributing to any voter's satisfaction. Such a "phantom" selection cannot be mapped back to an outcome of the original election, so the claimed correspondence between ILP solutions and EJR outcomes fails. A concrete example is n=2, ℓ=1, with voter 1 approving a and voter 2 approving b; the profile {1,2} is absent from the round, but the ILP allows selecting it. The fix is to restrict variables to the support of τ (or add explicit constraints x_{p,τ}=0 for absent p); with that fix the FPT claim in Corollary 7.3 would go through, but as written the theorem is not established.
minor comments (4)
  1. [Definition 2.5] The axiom is introduced as "extended justified representation (s-EJR)"; the acronym should be EJR to match Definitions 2.3–2.4 and the rest of the paper.
  2. [Section 2, discussion preceding Example 2.6] The two demand functions are both written as α, producing the self-referential inequality α(N′) ≥ α(N′); please use distinct symbols (e.g., α and α′) for the actual and alternative demand.
  3. [Theorem 6.1] The statement contains a typo: "togther" should read "together".
  4. [Section 5, first paragraph] The phrase "with respect to there parameters" should read "with respect to these parameters".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the verification-hardness results are proved by external reductions and the inherited demand function is a stated modeling choice, not a derived prediction.

full rationale

Finding: no significant circularity. The paper is a self-contained complexity-theoretic study. All hardness results (Theorems 3.1 and 3.2) are proved by explicit polynomial-time reductions from external NP-hard problems (CLIQUE, IS-3, Maximum Edge Biclique) and W[1]-hard Multicolored Clique, with full constructions and converse directions; they do not fit parameters, normalize data, or invoke the conclusion they are proving. The demand function alpha(N') = floor(beta(N')|N'|/n) is taken as a definition from Chandak et al. (2024); the paper does not derive it from the axioms, and its dependence on prior work is a modeling choice rather than a circular step. The paper explicitly addresses the natural alternative alpha' = min{beta, ell|N'|/n} and shows it is unsatisfiable even for JR (Example 2.6), so the inherited definition is not smuggled in as if it were forced. The cited Chandak et al. existence result for EJR outcomes is used only as background motivation for the choice of demand, not as a premise of the verification-hardness proofs. The appendix contains acknowledged gaps and conjectures (e.g., open problems in Section 4, the terse W[1]-hardness argument in Section B.2, and the semi-online impossibility in Section 7.3), but these are completeness and presentation limitations, not circular dependencies. No self-citation is load-bearing in the argument; the external benchmarks ground the reductions, and the classification has independent content beyond its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The paper relies on standard complexity-theoretic assumptions and on the domain definitions inherited from prior temporal voting work. No new entities are postulated.

assumptions (4)
  • standard math CLIQUE, IS-3, and MAXIMUM EDGE BICLIQUE are NP-hard; MULTICOLORED CLIQUE is W[1]-hard
    The coNP-hardness reductions in Theorems 3.1 and 3.2 reduce from these external problems; the W[1]-hardness in Theorem 5.4 reduces from Multicolored Clique. If these problems were tractable, the lower bounds would collapse.
  • domain assumption The temporal demand of a group is defined as alpha(N') = floor(beta(N') * |N'| / n), following Chandak et al. (2024)
    All results are stated for this specific demand function; Example 2.6 shows the alternative alpha' = min{beta, ell * |N'| / n} makes even JR unsatisfiable, so the complexity landscape depends on this modeling choice.
  • domain assumption Approval preferences are dichotomous and reported round by round
    The model uses approval sets per voter per round; the hardness and tractability results are for this representation, not for ranked preferences.
  • domain assumption The semi-online setting reveals preferences round by round with known horizon ell
    Proposition 7.4 concerns impossibility in this information model; different information models (offline, online) have different implications.

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Pith. "Pith review of Verifying Proportionality in Temporal Voting." pith.science (2026). https://pith.science/paper/6MDEXL5D

@misc{pith2026250205949,
  author       = {Pith},
  title        = {Pith review of: Verifying Proportionality in Temporal Voting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MDEXL5D}},
  note         = {Machine review of arXiv:2502.05949}
}
read the original abstract

We study a model of temporal voting where there is a fixed time horizon, and at each round the voters report their preferences over the available candidates and a single candidate is selected. Prior work has adapted popular notions of justified representation as well as voting rules that provide strong representation guarantees from the multiwinner election setting to this model. In our work, we focus on the complexity of verifying whether a given outcome offers proportional representation. We show that in the temporal setting verification is strictly harder than in multiwinner voting, but identify natural special cases that enable efficient algorithms.

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