Least Unpopularity in Matching and Voting

• Least unpopularity is defined in matching and voting models by minimizing defeat margins and last-place rankings to avoid strong majority defeats.
• Different formulations—margin-based, ratio-based, and disapproval-based—offer distinct measures for evaluating outcomes against opposition.
• Algorithmic and polyhedral methods reveal that computing least unpopularity is tractable within restricted classes but NP-hard in more general settings.

Searching arXiv for recent and foundational papers on least-unpopularity, popularity, and unpopularity-margin/factor notions. Least unpopularity is a family of selection principles in which an outcome is judged by the extent to which it can be defeated or broadly disliked, and the chosen outcome is one that eliminates strict majority defeats or minimizes a formal measure of defeat. In matching and branching theory, this appears as popularity, unpopularity margin, and unpopularity factor; in voting, it appears as the anti-plurality winner with the fewest last-place rankings. A common feature is that the criterion is comparative rather than cardinal: outcomes are evaluated against alternatives by votes, margins, ratios, or last-place counts, sometimes within a restricted admissible class such as maximum matchings or critical matchings (Kavitha et al., 2021, Kavitha et al., 2019, McCune et al., 25 Feb 2026).

1. Core formalizations

The term does not denote a single invariant across the literature. Instead, several mathematically precise notions serve the same role. In assignment and branching models, least unpopularity is typically expressed through head-to-head elections between feasible outcomes. In voting-theoretic models, it may instead mean minimizing the number of last-place rankings. These are not identical notions, but they share the objective of avoiding outcomes that are badly exposed to opposition.

Model	Quantity	Optimal condition
Assignment	$\mu(M) = \max_N \Delta(N,M)$	$M$ is popular iff $\mu(M)=0$
Branching	$\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$	$B$ is popular iff $\mu(B)=0$
Marriage/roommates	$u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$	$M$ is popular iff $u(M)\le 1$
Anti-plurality voting	$$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$	the least-unpopular candidate minimizes $M$0

In the one-sided assignment problem, for matchings $M$1, the vote margin is $M$2, and the unpopularity margin of $M$3 is $M$4. Popularity is exactly the zero-margin case. In branching problems, the same pattern appears with branchings in place of matchings. In the marriage and roommates problems, the emphasis shifts from a margin to a ratio: the unpopularity factor of a matching $M$5 is the maximum ratio $M$6 over competing matchings. In anti-plurality voting, least unpopularity is instead defined directly by last-place counts rather than by pairwise margins (Kavitha et al., 2021, Kavitha et al., 2019, Ruangwises et al., 2018, McCune et al., 25 Feb 2026).

A useful general distinction follows. Margin-based notions minimize the worst net loss, factor-based notions minimize the worst loss ratio, and disapproval-based notions minimize the breadth of explicit rejection. This suggests that “least unpopularity” is best understood as a design pattern rather than a single formula.

2. Margin and factor formulations in matching and branching

In assignment models with agents on one side and objects on the other, popularity is induced by agent comparisons between two matchings. If $M$7 denotes the partner of agent $M$8 in matching $M$9, then $\mu(M)=0$0 counts the agents that prefer $\mu(M)=0$1 to $\mu(M)=0$2, and $\mu(M)=0$3 measures the largest possible majority defeat of $\mu(M)=0$4. The LP formulation uses edge weights $\mu(M)=0$5, a primal assignment LP, and a dual LP; the optimal value of the primal equals $\mu(M)=0$6. The paper gives dual certificates both for popularity and for bounded margin: $\mu(M)=0$7 is popular iff there exists an optimal dual solution $\mu(M)=0$8 with $\mu(M)=0$9, $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$0, and $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$1, while $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$2 has unpopularity margin at most $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$3 iff there exists a feasible dual with $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$4 and slightly relaxed bounds on $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$5 (Kavitha et al., 2021).

In directed branching problems, each node compares its incoming edge across two branchings, with being a root treated as worst. The unpopularity margin is

$\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$6

and least unpopularity is the optimization problem $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$7. After adding a dummy root and reducing to $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$8-arborescences, popularity can be tested through a min-cost arborescence problem with costs $\mu(B) = \max_{B'}(\phi(B',B)-\phi(B,B'))$9. The dual side has a laminar certificate $B$0, and a key identity is $B$1. This gives a structural interpretation of least unpopularity: the larger the maximal dual certificate, the smaller the worst defeat margin (Kavitha et al., 2019).

The unpopularity factor gives a different robustness metric. In the marriage and roommates problems,

$B$2

with the stated edge-case conventions when the denominator vanishes. Popularity is recovered as the threshold $B$3. The same paper extends the definition to weighted voting by replacing $B$4 with a weighted count $B$5, yielding $B$6. Margin and factor are not interchangeable: one controls additive defeat severity, the other multiplicative severity (Ruangwises et al., 2018).

3. Least unpopularity within restricted admissible classes

A major line of work studies least unpopularity not over all feasible outcomes, but over a constrained admissible class. In bipartite graphs with strict rankings on both sides, a maximum matching $B$7 is a popular max-matching if no maximum matching is more popular than $B$8, equivalently if $B$9 for every maximum matching $\mu(B)=0$0. The paper explicitly interprets this as least unpopularity inside the class of maximum matchings: within $\mu(B)=0$1, the worst defeat margin is minimized to $\mu(B)=0$2, so a popular max-matching is a class-restricted Condorcet winner. The same work proves a compact extended formulation for the popular max-matching polytope via a surjective linear map from the stable matching polytope of an auxiliary instance $\mu(B)=0$3, implying that a min-cost popular max-matching can be computed in polynomial time. This contrasts with the min-cost popular matching problem over all matchings, which is NP-hard, and with min-cost Pareto-optimal matching or max-matching, which is also NP-hard (Kavitha, 2020).

An analogous restriction appears in many-to-many matching with two-sided lower quotas. There, a matching is critical if it minimizes total deficiency

$\mu(B)=0$4

Popularity is then evaluated only inside the set $\mu(B)=0$5 of critical matchings. The unpopularity margin within $\mu(B)=0$6 is

$\mu(B)=0$7

and a matching popular in $\mu(B)=0$8 satisfies $\mu(B)=0$9. The paper proves that such a matching always exists, gives a polynomial-time algorithm returning one of maximum cardinality, establishes a Rural Hospitals–type invariance for max-size popular critical matchings, and derives a $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$0 size guarantee relative to an unconstrained maximum-cardinality matching (Nasre et al., 2022).

In one-to-one object allocation, the admissible class can also be restricted implicitly by fairness requirements. A minimal envy matching is defined by a two-stage minimization of envy and has the operational characterization that every first house is matched to one of its first agents and, among such matchings, the number of agents who receive their first or second house is maximized. When a popular matching exists, it is equivalent to a minimal envy matching. When a popular matching does not exist, the paper defines a most popular matching as one that is popular on a largest subset of agents, and proves that every minimal envy matching is most popular. A plausible implication is that least unpopularity can be realized either by eliminating global defeats or by maximizing the size of the population on which no defeat occurs (Kondratev et al., 2019).

4. Relaxations when outright popularity fails

Universal existence often requires weakening the target notion. In general graphs with weak rankings, a semi-popular matching is one that does not lose a head-to-head election against a majority of matchings. If $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$1 is the set of all matchings and $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$2, semi-popularity is equivalent to

$u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$3

The Copeland score is

$u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$4

and every Copeland winner is semi-popular. Semi-popular matchings always exist, and the paper gives an FPRAS that, for any $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$5, returns with high probability a matching $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$6 such that $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$7 for at least a $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$8-fraction of all matchings. By contrast, computing a Copeland winner is NP-hard unless $u(M)=\max_{M'\neq M}\phi(M',M)/\phi(M,M')$9 (Kavitha et al., 2021).

For rankings compared by Kendall distance, the literature distinguishes weak and strong popularity. A ranking $M$0 is weakly popular if no alternative $M$1 is preferred by an absolute majority of voters; it is strongly popular if no alternative is preferred by a simple majority of non-abstaining voters. The paper establishes the inclusion chain

$M$2

and proves that weak and strong popularity coincide for $M$3 voters, while a separation exists at six voters. It also proves that any $M$4-sorted ranking is weakly popular, and that the search versions of $M$5-wurv, $M$6-surv, and $M$7-surv are NP-hard. This suggests that least unpopularity can be formalized either against all voters or only against non-abstainers, with materially different existence and complexity behavior (Kraiczy et al., 2021).

These relaxations address a common misconception. Least unpopularity is not always equivalent to finding a global Condorcet-style winner. In several models, such winners may fail to exist, and the operative substitute becomes majority resilience against most opponents, maximization of a Copeland score, or protection against the strongest admissible type of defeat.

5. Disapproval-based least unpopularity in voting and elimination

In ranked-choice voting theory, the least-unpopular candidate is the anti-plurality winner. With $M$8 candidates and $M$9 voters,

$u(M)\le 1$0

and the anti-plurality rule elects the candidate with the fewest last-place votes. The 2026 analysis of IRV studies the mirror question: when does IRV elect the most unpopular candidate, defined in one of its four senses as the candidate with the largest $u(M)\le 1$1? The paper proves that IRV can elect the candidate with the most last-place votes, but under complete, single-peaked preferences it cannot. For three candidates with complete ballots, the exact limiting probability that IRV elects the most-last-place candidate is $u(M)\le 1$2 under impartial anonymous culture and approximately $u(M)\le 1$3 under impartial culture. In Monte Carlo spatial simulations with complete ballots, the probability is $u(M)\le 1$4 in one-dimensional single-peaked models and ranges from about $u(M)\le 1$5 to $u(M)\le 1$6 in two-dimensional models, increasing with polarization. In empirical data completed to three ranks, the frequency is $u(M)\le 1$7 in Australia, $u(M)\le 1$8 in Scotland, $u(M)\le 1$9 in Scot_condensed, and $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$0 in the USA (McCune et al., 25 Feb 2026).

A related metric-space literature studies elimination of a single candidate using only favorite-candidate reports. There, for a candidate $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$1, the randomized Proportionality mechanism chooses the committee $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$2 with probability proportional to

$$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$3

that is, proportional to the number of voters who do not name $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$4 as favorite. The paper explicitly identifies this as directly connected to “least unpopularity.” It also shows that naive least-popular elimination can be arbitrarily poor, whereas Proportionality is strategy-proof and attains distortion at most $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$5 when $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$6 and at most $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$7 in simplex metrics; on the real line, the Left-or-Right mechanism achieves distortion at most $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$8 (Chen et al., 2019).

The voting literature therefore uses least unpopularity in a distinctly disapproval-based sense. Rather than minimizing worst pairwise defeat, it minimizes or probabilistically targets breadth of non-support, typically measured by last-place votes or by the number of voters who do not support a candidate.

6. Polyhedral methods, algorithms, and complexity

A recurrent structural theme is dual certification. In one-sided assignments, the level-raising algorithm builds a subgraph $$L_i = \sum_{v=1}^n \mathbf{1}\{\text{rank}_v(i)=m\}$$9, raises object levels, and either finds a perfect matching or certifies that no popular assignment exists; its running time is $M$00. For the $M$01-unpopularity-margin problem, the same paper gives an $M$02 algorithm, written as $M$03, and proves NP-completeness under strict rankings together with $M$04-hardness under weak rankings. The monotonicity of the predicate “there exists an assignment with $M$05” yields parametric search for the least possible margin $M$06 (Kavitha et al., 2021).

In branchings, the dual side is again central. Popularity is characterized by laminar dual certificates, and under weak rankings the MinMargin algorithm computes an arborescence of minimum unpopularity margin in polynomial time. The same paper shows that the popular branching polytope has an $M$07-size description in the original space, admits a compact extended formulation, and that minimum-margin branching becomes NP-hard under arbitrary partial order preferences. Under strict rankings, a different relaxation appears: there always exists a branching with unpopularity factor at most $M$08, and this guarantee is tight (Kavitha et al., 2019).

Polyhedral tractability and hardness contrasts are especially sharp for restricted feasible sets. Popular max-matchings admit a compact extension through a stable-matching instance $M$09, so min-cost popular max-matching is polynomial-time solvable. Yet the min-cost popular matching problem over all matchings is NP-hard, and min-cost Pareto-optimal matching or max-matching is NP-hard as well. This is a direct illustration that least unpopularity may become tractable only after the admissible class is restricted in a structurally compatible way (Kavitha, 2020).

For ratio-based least unpopularity, the central result is exact evaluation rather than optimization. The unpopularity factor of a given matching can be computed in $M$10 time for the roommates problem and in $M$11 time for the marriage problem, with the same asymptotic bounds in the weighted setting. The reduction uses an auxiliary graph $M$12 and tests whether $M$13 by searching for a positive-weight perfect matching in the general case or a positive-weight directed cycle in the bipartite case (Ruangwises et al., 2018).

Taken together, these results show that least unpopularity is algorithmically heterogeneous. Zero-margin outcomes may be guaranteed in some classes and absent in others; exact optimization may be polynomial in one preference model and NP-hard in a nearby one; and the most useful certificate may be a dual vector, a laminar family, a compact extension, a level function, or a direct counting formula. The unifying idea is not a fixed algorithmic template, but a consistent objective: select an outcome whose exposure to opposition is as small as the model permits.