Approval-Based Committee Elections Overview

• [Approval-based committee (ABC) elections] are multiwinner voting systems where voters approve any subset of candidates, and a fixed-size committee is chosen based on these approvals, aiming to balance individual welfare, diversity, and proportional representation, with applications in parliamentary decisions, organizational committees, and participatory budgeting.
• Key approval-based committee (ABC) rules, such as AV, PAV, and CC, vary in their methods for maximizing approval support, representing voters, and computed through site of NP-completeness complexity, with the balance of global optimization and parallel computability.
• Participation incentives, resignation monotonicity, and real-world constraints like block chain validator selections, relational databases, and civic deliberation platforms, introduce complexity; resulting in trade-offs between computational efficiency, representation quality, and security, with methods addressing these challenges (e.g., sequential Phragmén, Maximus rule, fractional committees e become NP-hard).

Approval-based committee elections are multiwinner elections in which each voter approves an arbitrary subset of candidates and a fixed-size committee is selected from the candidate set. Formally, an election consists of voters $N$, candidates $C$, approval ballots $A_i\subseteq C$, and a target committee size $k$; a committee is $W\subseteq C$ with $|W|=k$. Approval-based committee (ABC) rules map this instance to one or more winning committees. The framework separates candidates into approved and disapproved alternatives but does not encode rankings, intensities, complementarities, indifference, or opposition. Its principal design tensions are between individual or majoritarian welfare, diversity, proportional representation, fairness of voting power, computational tractability, strategic robustness, and stability.

1. Formal model and rule families

For an election instance $(N,C,\mathcal A,k)$, voter $i$'s approval ballot is $A_i\subseteq C$, and the supporters of candidate $c$ are

$C$0

A committee $C$1 satisfies $C$2. An ABC rule may be resolute, returning one committee, or irresolute, returning a nonempty set of tied committees. Candidate and voter identities are ordinarily treated symmetrically through neutrality and anonymity, although deterministic tie-breaking can sacrifice neutrality.

The most general scoring-based framework assigns voter $C$3 a score depending on the number of approved committee members and, potentially, the ballot size:

$C$4

A committee maximizing this score is selected. ABC scoring rules include the important subclass of Thiele rules, for which the score depends only on $C$5:

$C$6

where $C$7 and $C$8 is nondecreasing. Multiwinner Approval Voting (AV), Proportional Approval Voting (PAV), and approval-based Chamberlin–Courant (CCAV) are Thiele rules:

$C$9

$A_i\subseteq C$0

and

$A_i\subseteq C$1

A second important subclass is ballot-size weighted approval voting (BSWAV), whose score is

$A_i\subseteq C$2

AV has $A_i\subseteq C$3, while Satisfaction Approval Voting (SAV) has $A_i\subseteq C$4. BSWAV is additive across candidates, whereas non-linear Thiele rules create diminishing or otherwise ballot-dependent marginal returns.

Sequential rules construct a committee one candidate at a time rather than globally maximizing a committee score. Sequential PAV adds the candidate with the greatest PAV marginal contribution; sequential CC adds the candidate representing the largest number of currently unrepresented voters; sequential Phragmén minimizes an evolving maximum voter load; and the Method of Equal Shares (MES) selects candidates through voter budgets. Sequentialization generally improves computational tractability but can introduce path dependence, participation failures, and intrinsic sequentiality.

ABC elections are used for parliamentary and organizational committees, expert shortlisting, finalist selection, group recommendation, collaborative filtering, search-result diversification, participatory budgeting, dynamic question selection, and validator selection. The simplicity of approval ballots makes them suitable for these applications, but their binary form cannot distinguish between a candidate who is actively disliked and one who is merely irrelevant (Lackner et al., 2020).

2. Welfare, diversity, and proportional representation

Approval Voting maximizes total approval support:

$A_i\subseteq C$5

It is polynomial-time computable, efficient, committee monotone, and strategyproof under fixed tie-breaking, but it is generally non-proportional: a large cohesive group can occupy nearly the entire committee.

CC maximizes the number of represented voters:

$A_i\subseteq C$6

It is equivalent to Maximum Coverage, is NP-hard to optimize, and emphasizes diversity rather than repeated representation. Its objective is monotone submodular, yielding a standard greedy approximation of at least

$A_i\subseteq C$7

PAV uses harmonic diminishing returns. A voter receiving $A_i\subseteq C$8 approved committee members contributes $A_i\subseteq C$9, so the marginal values of successive representatives are $k$0. This balances broad representation against the claims of large groups. PAV extends the D’Hondt apportionment method on party-list profiles, satisfies strong cohesive-group proportionality axioms, and achieves proportionality degree $k$1, which is optimal. Winner determination is NP-hard.

The standard proportionality hierarchy is

$k$2

A group $k$3 is $k$4-cohesive if

$k$5

and

$k$6

Justified representation (JR) requires every $k$7-cohesive group to contain at least one voter approving a committee member. Equivalently, no group of at least $k$8 voters who jointly approve some candidate may be entirely unrepresented.

Proportional justified representation (PJR) requires an $k$9-cohesive group to collectively approve at least $W\subseteq C$0 elected candidates:

$W\subseteq C$1

Extended justified representation (EJR) requires at least one voter in the group to approve at least $W\subseteq C$2 elected candidates:

$W\subseteq C$3

JR is always achievable in polynomial time, including through Greedy Approval Voting and threshold GAV. PAV satisfies both JR and EJR, whereas RAV, despite being a sequential approximation to PAV, can fail JR. Testing JR is polynomial-time, while testing EJR is coNP-complete. EJR can be computed for fixed $W\subseteq C$4 in time polynomial in $W\subseteq C$5 and $W\subseteq C$6, but no polynomial-time procedure for finding an EJR committee is known in general unless one solves an NP-hard PAV winner-determination problem (Aziz et al., 2014).

The distinction between welfare and power is central. PAV represents proportionality of satisfaction, while Phragmén and MES represent proportionality of decision-making power through loads, payments, or budgets. PJR and EJR do not by themselves prevent minority overrepresentation: a faction can satisfy underrepresentation criteria while still controlling too many seats.

3. Stability, core concepts, and maximin support

The core strengthens proportionality from cohesive-group requirements to arbitrary coalitional deviations. For a committee $W\subseteq C$7, an alternative $W\subseteq C$8 with $W\subseteq C$9 blocks $|W|=k$0 if at least

$|W|=k$1

voters strictly prefer $|W|=k$2 to $|W|=k$3, meaning

$|W|=k$4

A core committee satisfies

$|W|=k$5

for every $|W|=k$6. Core stability implies EJR, but EJR does not imply core stability because core deviations need not be cohesive: different coalition members may approve different candidates in $|W|=k$7.

The general nonemptiness of the approval-based core remains unresolved. Exact existence is known for several bounded regimes. Every PAV committee is core stable when $|W|=k$8; at $|W|=k$9, at least one PAV committee is core stable, although some PAV committees may fail the core. Core committees exist for every $(N,C,\mathcal A,k)$0 when $(N,C,\mathcal A,k)$1, using a recursive PAV-based construction and computer-assisted linear-programming certificates (Peters, 30 Jan 2025). A separate affine-monoid argument proves core nonemptiness for elections with at most five voters, and more generally with at most five distinct voter types; the proof uses fractional-to-integral rounding that preserves each voter’s utility up to floors (Becker et al., 7 May 2026). Deterministic Hare-core nonemptiness is also established for nine-seat elections, extending the known result to every committee size $(N,C,\mathcal A,k)$2 (Fang, 3 Sep 2026).

Approximate stability provides a general algorithmic substitute. A committee $(N,C,\mathcal A,k)$3 is $(N,C,\mathcal A,k)$4-approximately stable if every alternative committee $(N,C,\mathcal A,k)$5 is strictly preferred to $(N,C,\mathcal A,k)$6 by fewer than

$(N,C,\mathcal A,k)$7

Every ABC election admits a randomized polynomial-time $(N,C,\mathcal A,k)$8-approximately stable committee. The construction combines a fractional Lindahl equilibrium, strongly Rayleigh rounding, lower-tail concentration, greedy augmentation, and recursion. Its internal asymptotic factor is approximately $(N,C,\mathcal A,k)$9, with the final $i$0 factor accounting for integer-rounding effects (Gao et al., 31 Jul 2025).

For applications concerned with preventing overrepresentation, maximin support assigns voter strength to approved elected candidates. For a support allocation $i$1,

$i$2

The maximin-support problem chooses a committee and allocation maximizing the least-supported elected candidate. Equivalently,

$i$3

This objective protects against a faction obtaining a block of seats with little collective backing. It is NP-hard to optimize and is NP-hard to approximate within a factor better than $i$4 for every constant $i$5. The Phragmms rule combines Phragmén-style construction with balanced support allocations, satisfies PJR, and gives a $i$6-approximation to maximin support. Its certificate can be verified in $i$7 time, making it suitable for verifiable validator selection (Cevallos et al., 2020).

4. Computational complexity and algorithmic structure

ABC rules exhibit a sharp division between additive, globally optimized, and sequential procedures. AV and SAV are polynomial-time because their objectives decompose candidate by candidate. PAV, CC, Monroe, Minimax Approval Voting, and most non-linear weighted PAV rules have NP-hard winner determination. Sequential PAV, sequential CC, sequential Phragmén, Greedy Monroe, and MES are polynomial-time computable, but their sequential state updates can be inherently sequential in the parallel-complexity sense.

Winner determination for sequential CC, sequential PAV, reverse sequential CC, reverse sequential PAV, sequential Phragmén, Greedy Monroe, and MES is $i$8-hard. Under the standard assumption $i$9, these rules cannot generally be computed in polylogarithmic parallel time with polynomially many processors. The reductions encode sequential processes such as Ordered Vertices Remaining and Lexicographically First Maximal Independent Set. By contrast, AV and SAV are computable in deterministic logarithmic space and are therefore efficiently parallelizable. CC becomes parallelizable on single-peaked or single-crossing approval profiles, where approval sets or supporter sets are intervals (Fitzsimmons et al., 25 Jan 2025).

Characterization results clarify the structure of scoring rules. In the standard set-valued ABC model, a rule is a Thiele rule if and only if it satisfies anonymity, neutrality, consistency, continuity, and independence of losers. A rule is BSWAV if and only if it satisfies anonymity, neutrality, consistency, continuity, choice-set convexity, and weak efficiency. These results distinguish global additive scoring from sequential or non-additive procedures and provide axiomatic characterizations of AV, PAV, and SAV (Dong et al., 2023).

Incomplete information creates additional complexity. In models where voters partition candidates into approved, disapproved, and unknown categories, possible and necessary winning committees and membership problems depend strongly on whether unknown candidates are unconstrained, linearly ordered, or organized by a partial order. In a two-set approved/unknown model, every candidate can be completed as approved by every voter, making possible-winner and possible-membership questions trivial. PAV complexity remains open in the 3VA model but is hard in linear and poset models (Imber et al., 2021).

When approval information must be acquired through queries, adaptivity is significant. Non-adaptive information collection can require $A_i\subseteq C$0 queries to guarantee JR, with stronger constructions yielding $A_i\subseteq C$1 lower bounds for suitable committee sizes. Adaptive PAV-based procedures can achieve exact or approximate EJR and average-satisfaction guarantees using substantially fewer queries. Incomplete-vote settings for civic deliberation use adaptive query routing to estimate PAV marginal gains and select representative comments from sparse feedback (Halpern et al., 2022).

Online arrival creates a different constraint: candidates appear sequentially, and each accept/reject decision is irrevocable. Exact PJR is achievable through a Greedy Budgeting Rule, but exact EJR is impossible for deterministic online procedures. The Online Greedy Cohesive Algorithm achieves $A_i\subseteq C$2-EJR, where

$A_i\subseteq C$3

and this harmonic loss is essentially optimal. A polynomial-time variant achieves $A_i\subseteq C$4-EJR (Do et al., 2022).

5. Participation, monotonicity, and dynamic candidate sets

Participation asks whether a voter can benefit by abstaining. For committee preferences based on approval counts, voter $A_i\subseteq C$5 prefers $A_i\subseteq C$6 to $A_i\subseteq C$7 when

$A_i\subseteq C$8

An ABC rule satisfies participation if abstention never strictly improves the outcome; group participation prohibits beneficial abstention by any coalition.

All ABC scoring rules satisfy group participation, including AV, PAV, CCAV, and ballot-size weighted rules. The proof follows from additive score comparisons: if abstaining voters weakly prefer the post-abstention committee, removing their contributions cannot turn that committee into a winner when it was not previously one.

Sequential rules behave differently. Every sequential Thiele rule except sequential AV, as well as sequential Phragmén and MES, can fail participation for $A_i\subseteq C$9. A voter may receive one approved committee member when participating but $c$0 approved members after abstaining. Sequential Thiele rules and sequential Phragmén nevertheless satisfy participation for voters who approve none of the elected candidates. Sequential rules, sequential Phragmén, and MES satisfy participation on laminar profiles, where candidate-support sets are nested or disjoint. Determining whether abstention is beneficial is NP-hard for the principal sequential Thiele rules, MES, and sequential Phragmén (Bullinger et al., 2023).

Resignation monotonicity concerns elected candidates who leave after the election. If $c$1 resigns, the rerun must preserve every member of $c$2, replacing only the vacant seats. AV satisfies resignation monotonicity, but every non-linear Thiele method, including PAV and CC, fails it in general. More fundamentally, no voting rule can satisfy both resignation monotonicity and JR. Integral proportional representation can require replacing surviving winners to restore representation after resignations.

Fractional committees avoid this incompatibility. A fractional committee $c$3 with $c$4 can redistribute the probability mass of resigned candidates while weakly increasing the selection probability of every surviving candidate. Maximum-flow representations yield fractional committees satisfying Group Resource Proportionality and resignation monotonicity.

The integral Maximum Payment Rule (MPR) maximizes the total voter payment that can be routed through selected candidates. It satisfies PJR+ and a relaxed form of resignation monotonicity: after resignations, there exists a replacement committee preserving all surviving winners and satisfying PJR+. MPR is NP-hard in general but tractable on interval domains, for fixed numbers of voters or candidates, and when $c$5. It is closely connected to Monroe’s rule and induces the largest-remainder method on party-list profiles (Oh et al., 6 Aug 2026).

6. Applications, empirical evidence, and contextual constraints

Approval-based committee elections have been deployed or studied in blockchain validator selection, civic deliberation, search diversification, recommendation, and structured selection problems.

In Nominated Proof-of-Stake, nominators approve validator candidates, stake acts as weighted voting power, and a committee of validators participates in consensus. Polkadot elections contain approximately one thousand candidates and tens of thousands of weighted voters, with a committee of approximately 300 validators. Sequential Phragmén is used operationally. The security objective is not only to avoid minority underrepresentation but also to prevent a minority faction from obtaining a large block of validators with little backing stake.

A large-scale empirical study of 496 Polkadot elections compared AV, SAV, sequential PAV, sequential Phragmén, MES, and Phragmms. The proportional rules produced highly similar committees and substantially outperformed AV and SAV on representation, priceability, maximin support, and attack-oriented measures. AV and SAV frequently generated JR and EJR+ violations and were substantially cheaper to manipulate through coordinated attacks. Sequential Phragmén, MES, and Phragmms consistently produced priceable committees, while AV and SAV were far from priceable. Replacing one-third of the validator committee required approximately 14% of total stake under AV, 24% under SAV, and approximately 40% under Phragmms and sequential Phragmén (Boehmer et al., 2023).

Approval-based committee rules can also be combined with contextual constraints. A relational database may describe candidate attributes, affiliations, conflicts, topics, languages, or other relationships. A virtual relation $c$6 represents committee membership. Tuple-generating dependencies express representation requirements, while denial constraints express conflicts. The resulting optimization problem is

$c$7

subject to the database constraints being satisfied.

Even AV becomes NP-hard under such constraints: tuple-generating dependencies can encode Set Cover and 3-Dimensional Matching, while denial constraints can encode Maximum Independent Set. Key constraints yield tractable special cases, including greedy algorithms and minimum-cost maximum-flow formulations. A general mixed-integer programming implementation supports arbitrary ABC scoring rules, tuple-generating dependencies, and denial constraints, with voter grouping, score pruning, and clique or hyperclique contraction (Yona et al., 27 Jan 2025).

In civic deliberation platforms such as Polis, participants typically approve only a small fraction of available comments. Adaptive PAV-based methods can acquire information selectively and produce more representative committees than approval-score aggregation. In diversity-oriented applications, the CC objective is Maximum Coverage. Adaptive greedy and local-search methods approach the $c$8 approximation threshold with high probability under incomplete information, while non-adaptive algorithms face quadratic query lower bounds. With inaccurate responses, repeated querying and majority reconstruction achieve a $c$9 approximation with query complexity $C$00 for fixed noise rate (Lindeboom et al., 12 Jun 2025).

Uncertainty about approval ballots also changes the computational problem. Under candidate-wise independent approval probabilities, expected social welfare remains easy to optimize by selecting candidates with the highest expected approval scores. By contrast, selecting the committee most likely to be exactly welfare-maximizing is NP-hard in joint-probability and lottery models, and welfare-optimality probability is $C$01-complete in the lottery model. In the candidate-probability and 3VA models, welfare distributions and welfare-maximizing probabilities are polynomial-time computable through Poisson-binomial dynamic programming (Aziz et al., 2 Mar 2025).

The principal unresolved issues concern the simultaneous achievement of proportionality, stability, efficiency, strategy resistance, monotonicity, scalability, and dynamic robustness. No rule dominates across these dimensions. AV is additive, fast, parallelizable, and strategyproof but majoritarian; PAV offers strong welfare-based proportionality but is NP-hard, non-monotone with respect to committee size, and vulnerable to some dynamic effects; MES and Phragmén provide power-based proportionality but can be sequential and computationally non-parallelizable; maximin-support methods address overrepresentation and security at the cost of more complex optimization; and core stability remains incompletely understood beyond bounded regimes.