Approval-Based Committee Selection

• Approval-Based Committee Selection is defined as choosing a fixed-size committee from a candidate set using voters' approval ballots, emphasizing fairness and proportional representation.
• The literature examines various scoring and sequential rules such as AV, PAV, SAV, and CCAV to balance utilitarian scores, proportionality, stability, and computational tractability.
• Recent research highlights existence theorems and algorithmic strategies, including greedy methods, fractional formulations, and randomized approaches for achieving core stability.

Approval-Based Committee Selection (ABCS), also called approval-based committee voting or approval-based committee elections, studies the choice of a fixed-size committee from a candidate set on the basis of approval ballots. In the standard model, each voter approves a subset of candidates, and a committee is evaluated through the number of approved winners it contains for each voter; the literature then asks how different rules trade off utilitarian score, proportional representation, stability, and computational tractability. The area includes scoring rules such as AV, PAV, SAV, and CCAV, sequential rules, fractional and lottery-based formulations, and strong fairness notions such as the core (Peters, 30 Jan 2025, Becker et al., 7 May 2026).

1. Formal model

The canonical ABCS instance consists of a candidate set $C$ with $|C|=m$, a voter set $N$ with $|N|=n$, approval ballots $A_i \subseteq C$ for each voter $i$, and a target committee size $k$. A committee is a subset $W \subseteq C$ with $|W|=k$. The standard voter-level utility is

$u_i(W)=|W \cap A_i|,$

that is, the number of approved committee members. Some treatments replace the explicit voter multiset by a distribution $|C|=m$0 over approval sets, with $|C|=m$1 the fraction of voters casting ballot $|C|=m$2 (Peters, 30 Jan 2025).

A second formal layer uses fractional committees. Here, a fractional committee is a vector $|C|=m$3 with $|C|=m$4, and its approval utility is linear: $|C|=m$5 This formulation is central in recent work on the core, weighted voters, and supply caps, where candidates may also be aggregated into approval types $|C|=m$6 and feasibility is expressed in type space rather than candidate space (Becker et al., 7 May 2026).

This standardization is one reason the ABCS literature connects easily to social choice, combinatorial optimization, cooperative game theory, online algorithms, and market-equilibrium methods. The same base model supports scoring rules, proportionality axioms, exact and approximate stability, and structured-domain algorithms.

2. Rule families and axiomatic structure

A broad class of ABC rules are ABC scoring rules. These are defined by a scoring function $|C|=m$7, where $|C|=m$8 is the number of approved committee members obtained by a voter and $|C|=m$9 is the ballot size. The score of a committee $N$0 at profile $N$1 is

$N$2

and winners are the score-maximizing committees. Two prominent subclasses are Thiele rules, where the score depends only on $N$3, and ballot size weighted approval voting (BSWAV), where the score is linear in $N$4 with ballot-size-dependent weights. In the standard set-of-winners model, Thiele rules are characterized by anonymity, neutrality, consistency, continuity, and independence of losers; BSWAV rules are characterized by anonymity, neutrality, consistency, continuity, choice set convexity, and weak efficiency (Dong et al., 2023).

Within the Thiele class, several canonical rules recur throughout the literature. Approval Voting (AV) uses $N$5. Proportional Approval Voting (PAV) uses

$N$6

Chamberlin–Courant Approval Voting (CCAV) uses $N$7 if $N$8 and $N$9 otherwise. Satisfaction Approval Voting (SAV) uses $|N|=n$0. These rules differ in whether they reward repeated representation of the same voter linearly, harmonically, once only, or relative to ballot size (Dong et al., 2023).

A separate but closely related strand studies sequential valuation rules. These build a committee greedily by repeatedly adding a candidate with maximum marginal gain. The relevant axiomatic notion is consistent committee monotonicity (CCM). An ABC rule is a step-dependent sequential scoring rule iff it is proper and consistently committee monotone. Adding independence of losers yields the step-dependent sequential Thiele rules, and adding committee separability yields the sequential Thiele rules. This framework characterizes seqAV, seqPAV, and seqCCAV inside the broader sequential family (Dong et al., 2023).

These characterizations show that ABCS is not merely a catalog of named rules. The main rule classes arise from explicit structural axioms: additive separability for scoring rules, CCM for sequential rules, and specific ballot-size or marginal-gain constraints for finer subclasses.

3. Representation, proportionality, and stability

The basic proportionality language of ABCS is built from cohesive groups. For $|N|=n$1, a group $|N|=n$2 is $|N|=n$3-cohesive if $|N|=n$4 and $|N|=n$5. A committee satisfies JR if every $|N|=n$6-cohesive group gets at least one approved representative; PJR if every $|N|=n$7-cohesive group gets at least $|N|=n$8 winners in the union of its approvals; and EJR if every $|N|=n$9-cohesive group contains some voter with at least $A_i \subseteq C$0 approved winners. The implication chain is

$A_i \subseteq C$1

The approval-core literature places the core strictly above EJR in strength (Aziz et al., 2014, Brill et al., 2021, Peters, 30 Jan 2025).

The core is a stability notion based on proportional deviations. In the Hare-core formulation, a committee $A_i \subseteq C$2 is in the core iff there do not exist a nonempty coalition $A_i \subseteq C$3 and a committee $A_i \subseteq C$4 with $A_i \subseteq C$5 such that every voter in $A_i \subseteq C$6 strictly prefers $A_i \subseteq C$7 to $A_i \subseteq C$8: $A_i \subseteq C$9 A fractional version allows $i$0 instead of integral committees. For approval utilities, the fractional core is non-empty; the central question is when this can be rounded to an integral core committee (Becker et al., 7 May 2026).

A related line studies fairness via stability of lotteries. A lottery $i$1 over committees of size $i$2 is stable if for every deviating set $i$3,

$i$4

where $i$5 is the number of voters who strictly prefer $i$6 to $i$7. This generalizes deterministic core-style blocking to randomized outcomes and guarantees stable lotteries in the Approval Set model (Cheng et al., 2019).

ABCS also has an explicitly individual notion, Individual Representation (IR). For voter $i$8,

$i$9

and a committee satisfies IR if $k$0 for all $k$1. IR implies EJR, but it is incompatible with core stability in some instances; this sharpens the difference between group-level and per-voter fairness (Brill et al., 2021).

4. Existence theorems and constructive results

Several central existence questions in ABCS now have partial positive resolutions. For JR, existence is straightforward: Greedy Approval Voting and its threshold variant always output JR committees, and checking JR is polynomial-time. For EJR, PAV satisfies EJR, and a polynomial-time local-search rule based on approximately maximizing the PAV score returns committees that guarantee $k$2 for every $k$3-cohesive group, which implies EJR (Aziz et al., 2014, Skowron et al., 2017).

The exact core remained open for the general approval setting, but recent work has established several sharp frontiers. The core is always non-empty when $k$4, and also whenever $k$5. More precisely, every PAV-selected committee is in the core for $k$6, and for $k$7 at least one PAV winner is core-stable. The few-candidates result uses a recursive variant of PAV and computer-aided LP proofs (Peters, 30 Jan 2025).

A different frontier is the number of voters or voter types. For approval-based committee elections with $k$8, there always exists an integral committee in the core. The same result extends to weighted voters and to profiles with at most five distinct approval sets. The proof passes from the non-empty fractional core to the integral core by a deterministic rounding lemma preserving utility floors,

$k$9

and identifies an affine monoid whose normality holds exactly for $W \subseteq C$0. In these cases, a core committee can be computed in polynomial time (Becker et al., 7 May 2026).

Exact stability is still not known in full generality, so the literature also studies randomized and approximate substitutes. Stable lotteries always exist in the Approval Set model, and an $W \subseteq C$1-approximately $W \subseteq C$2-stable lottery can be computed in $W \subseteq C$3 time (Cheng et al., 2019). For deterministic committees, a randomized polynomial-time algorithm computes a $W \subseteq C$4-approximately stable committee using a Lindahl equilibrium and sampling from a strongly Rayleigh distribution associated with it (Gao et al., 31 Jul 2025).

These results collectively move ABCS from existence-by-axiom toward structural existence theorems with explicit algorithms: local search for EJR, recursive PAV for small $W \subseteq C$5, affine-monoid rounding for at most five types, and equilibrium-plus-rounding methods for approximate stability.

5. Algorithmic complexity and computation

Winner determination in ABCS spans the full range from trivial to intractable. AV is polynomial-time, but winner determination is NP-hard for Thiele rules other than AV, including PAV and CCAV (Dong et al., 2023). Exact PAV is NP-hard in general, and checking whether a committee is core-stable is coNP-complete (Peters, 30 Jan 2025). The outlier model adds another layer: with outliers, approval, net approval, and minisum variants become NP-complete, and approval and net approval admit no $W \subseteq C$6-factor approximation unless $W \subseteq C$7 (Dey et al., 2015).

The complexity picture also changes under representation and fairness constraints. Deciding whether an IR committee exists is NP-hard, and computing the entitlement threshold $W \subseteq C$8 is NP-complete (Brill et al., 2021). Adding database-style context constraints likewise raises difficulty: tuple-generating dependencies and denial constraints make even AV NP-hard in general, though the presence of key constraints yields tractable cases and admits a Mixed Integer Programming formulation that supports arbitrary ABC scoring rules (Yona et al., 27 Jan 2025).

Parallel computation introduces a different barrier. Computing winning committees for seq-CC, seq-PAV, rev-seq-CC, rev-seq-PAV, sequential Phragmén, Greedy Monroe, MES, and MES+seq-P is P-hard, so these rules are not parallelizable unless $W \subseteq C$9. By contrast, AV and SAV are in $|W|=k$0, and approval-based Chamberlin–Courant becomes parallelizable on single-peaked or single-crossing profiles, where SP-CC and SC-CC lie in OptL (Fitzsimmons et al., 25 Jan 2025).

This landscape makes computational structure a first-order design variable in ABCS. The choice of rule is often inseparable from the admissible domain, the availability of approximation, and whether one needs sequential, distributed, or certifiable computation.

6. Structured domains and major extensions

Several extensions of ABCS preserve the approval-ballot core while changing the environment. In online approval committee elections, candidates arrive sequentially and must be accepted or rejected irrevocably. Under random-order arrival, a secretary-style algorithm for monotone submodular committee scores guarantees

$|W|=k$1

For representation, the Greedy Budgeting Rule satisfies PJR, OGCA satisfies $|W|=k$2-EJR, no online rule can satisfy $|W|=k$3-EJR, and SGBR gives a polynomial-time $|W|=k$4-EJR guarantee (Do et al., 2022).

Under uncertain approvals, the literature studies Joint Probability, Lottery, Candidate Probability, and Three-Valued Approval models. One line analyzes social-welfare maximization under uncertainty, including the problems IsPossSWM, IsNecSWM, SWM-Prob, SW-Dist, and MaxSWM; another studies the probability that a fixed committee satisfies JR, PJR, or EJR. The computational behavior varies sharply by model: for example, SWM-Prob is in $|W|=k$5 in the Candidate Probability model but $|W|=k$6-complete in the Lottery model, and JR-Probability is $|W|=k$7-complete in 3VA while IsNecJR is polynomial-time in all four models (Aziz et al., 2 Mar 2025, Aziz et al., 2024).

Structured preference domains alter individual guarantees as well. For IR, the literature shows a strong contrast between voter interval and candidate interval profiles: under voter interval preferences there is a polynomial-time algorithm returning a $|W|=k$8-IR committee, whereas candidate interval profiles can admit no $|W|=k$9-IR committee for any $u_i(W)=|W \cap A_i|,$0 and $u_i(W)=|W \cap A_i|,$1 (Brill et al., 2021). This suggests that domain restrictions can radically change attainability.

A separate extension enriches the ballot language itself. In the compatibility principle model, voters approve not only candidates but coalitions of candidates. This produces a TU cooperative game $u_i(W)=|W \cap A_i|,$2, where $u_i(W)=|W \cap A_i|,$3 is the number of voters approving coalition $u_i(W)=|W \cap A_i|,$4, and candidates are scored by their Shapley values; the top-$u_i(W)=|W \cap A_i|,$5 Shapley scores define the committee. In the singleton-only special case, the rule collapses to ordinary approval ranking (Dutta et al., 2023).

Across these extensions, ABCS remains recognizable—approval ballots, fixed-size committees, and committee-level fairness remain central—but the surrounding mathematical apparatus shifts to secretary algorithms, stochastic optimization, interval structure, database dependencies, or cooperative game theory. That breadth is now a defining feature of the field rather than an exception.