Popular Matchings

• Popular matchings are defined by a majority vote where vertices compare their partners, replacing local blocking-pair logic with a global criterion.
• The concept spans various models such as one-sided house allocation, two-sided marriage, and roommates, each with tailored algorithmic approaches.
• Variants like dominant, popular max-, and robust popular matchings highlight clear tractability boundaries and complex decision problems in matching theory.

Popular matchings are a voting-based solution concept for matching problems: a matching is popular if no alternative matching defeats it in a head-to-head election in which vertices compare the partners they receive under the two outcomes. In this sense, popularity replaces the local blocking-pair logic of stability with a global majority criterion. The notion has been studied in one-sided house allocation, two-sided marriage markets, and non-bipartite roommates models, and has since been extended to weighted voters, robust multi-instance comparisons, capacitated many-to-many settings, and committee-style “popular winning sets” when no single popular matching exists (Cseh et al., 2018, Connor et al., 29 Sep 2025).

1. Formal model and voting semantics

In the basic bipartite setting, one is given a graph $G=(A\cup B,E)$, where acceptable pairs are edges and each vertex has a strict preference order over its neighbors. For a matching $M$, $M(v)$ denotes the partner of $v$, with $M(v)=\bot$ if $v$ is unmatched. A vertex $v$ prefers matching $M$ to matching $M'$ if $M(v)$ is preferred to $M$0, with being matched preferred to being unmatched. Writing

$M$1

the unweighted popularity margin is

$M$2

A matching $M$3 is more popular than $M$4 if $M$5, and $M$6 is popular if $M$7 for every matching $M$8. In the weighted-voter generalization, each vertex has weight $M$9, and the margin becomes

$M(v)$0

The resulting decision problem asks whether there exists a matching $M(v)$1 such that $M(v)$2 for all matchings $M(v)$3 (Heeger et al., 2021).

This framework subsumes several classical models. In one-sided house allocation, only agents vote and houses can be modeled as weight-0 vertices. In two-sided marriage-style models, both sides vote. In roommates settings the graph is non-bipartite, but the same pairwise-election interpretation applies. The common core is that popularity is a Condorcet-style property over the space of matchings rather than a local feasibility condition on individual edges (Heeger et al., 2021).

2. Foundational characterizations

The first complete structural characterization arose in one-sided house allocation. For each agent $M(v)$4, let $M(v)$5 be its first choice and let $M(v)$6 be its best-ranked neighbor that is not anyone’s first choice, or $M(v)$7 if none exists. In the Abraham–Irving–Kavitha–Mehlhorn characterization, a matching is popular if and only if every house that is some $M(v)$8 is matched to an agent whose first choice is that house, and every agent is matched either to $M(v)$9 or to $v$0. This yields an $v$1-time algorithm for maximum-size popular matchings or a proof of non-existence in the one-sided setting (Heeger et al., 2021).

A more general line of work expresses popularity through linear programming duality. For a fixed matching $v$2, one augments the graph by adding a loop $v$3 at each vertex, interprets loops as being unmatched, and defines edge weights from local vote comparisons against $v$4. In the unweighted bipartite case, $v$5 is popular if and only if there exists a witness vector $v$6 such that $v$7, every edge constraint satisfies $v$8, and every loop constraint is satisfied. In the weighted setting, the same construction leads to a rational witness $v$9 with $M(v)=\bot$0, edge inequalities $M(v)=\bot$1, lower bounds $M(v)=\bot$2 on unmatched vertices, and $M(v)=\bot$3 in general. These witness theorems are the main polyhedral certificates used in both algorithms and hardness reductions (Heeger et al., 2021).

In one-sided object allocation, popularity also admits a fairness interpretation. A minimal envy matching is one that first minimizes the total number of envying agents and then minimizes envy in the reduced problem obtained after removing non-envying agents and their houses. This is equivalent to requiring that each first house be allocated to a first agent and, among such matchings, maximizing the number of agents who receive either their first or their second house. Whenever a popular matching exists, the set of minimal envy matchings coincides with the set of popular matchings. In the same model, global popularity is equivalent to local popularity against exchanges by coalitions of size at most 3, and arbitrary matchings are path-connected to popular matchings through sequences of such local majority-improving exchanges (Kondratev et al., 2019).

3. Stable, dominant, and related subclasses

Stable matchings occupy one extreme of the popular-matching landscape. In the classical strict-preference settings, every stable matching is popular, and stable matchings are minimum-size popular matchings. At the other extreme are dominant matchings: a popular matching $M(v)=\bot$4 is dominant if it strictly defeats every larger matching. In bipartite graphs, a popular matching is dominant if and only if the graph $M(v)=\bot$5, obtained by deleting all $M(v)=\bot$6-labeled edges relative to $M(v)=\bot$7, contains no $M(v)=\bot$8-augmenting path. Dominant matchings therefore form a max-size subclass of popular matchings, while stable matchings form the min-size subclass (Cseh et al., 2018).

Between these two extremes lies the notion of a popular max-matching. Here the comparison is restricted to maximum-cardinality matchings: a maximum matching $M(v)=\bot$9 is a popular max-matching if no other maximum matching is more popular. Such matchings always exist in bipartite graphs with strict preferences. Their polyhedral structure is unusually clean: the popular max-matching polytope admits a compact extended formulation obtained by projecting the stable matching polytope of an auxiliary instance $v$0, and this yields a polynomial-time algorithm for computing a minimum-cost popular max-matching (Kavitha, 2020).

For general, possibly non-bipartite graphs, a stricter class called strongly dominant matchings isolates the tractable part of dominance. A matching $v$1 is strongly dominant if there exists a partition $v$2 of the vertices such that $v$3, all vertices in $v$4 are matched, every $v$5 edge lies inside $v$6, and every edge inside $v$7 is $v$8. This is equivalent to the existence of a witness vector $v$9 with $v$0 on matched vertices and $v$1 on unmatched vertices. Strongly dominant matchings can be recognized in linear time via a reduction to a stable matching problem in an auxiliary graph (Faenza et al., 2018).

4. Complexity landscape and tractability boundaries

The existence and optimization complexity of popular matchings is highly sensitive to the model. In two-sided bipartite graphs with all vertex weights equal to 1, popular matchings always exist because every stable matching is popular. As soon as vertex weights vary, however, existence can fail and the decision problem becomes hard. Popular Matching with Weighted Voters is NP-complete even when all but 14 vertices have weight 1 and the remaining weights belong to $v$2. For the asymmetric regime $v$3 for all $v$4 and $v$5 for all $v$6, existence is NP-complete for every constant $v$7, while for $v$8 one can compute a maximum-cardinality popular matching, or prove non-existence, in $v$9 time; the interval $M$0 remains open (Heeger et al., 2021).

In roommates-style complete graphs, parity alone changes the picture. For a complete graph on an odd number of vertices, every popular matching is stable, so existence reduces to the stable roommates problem and is polynomial-time decidable. For a complete graph on an even number of vertices, deciding whether a popular matching exists is NP-complete (Cseh et al., 2018).

One-sided ties on one side of a bipartite graph define another sharp boundary. If applicants in $M$1 have strict preferences and every $M$2 places all its neighbors in a single tie, then the popular matching problem is solvable in $M$3 time. If vertices in $M$4 are allowed either strict lists or a single tie over all neighbors, the problem becomes NP-complete even under strong degree restrictions (Cseh et al., 2016).

These results align with a broader tractability thesis for bipartite popular matchings: stable and dominant matchings are the tractable extremal subclasses, while intermediate popular matchings are hard to detect. In particular, deciding whether a bipartite instance admits a popular matching that is neither stable nor dominant is NP-complete. In non-bipartite graphs the situation is harsher: deciding whether a popular matching exists is NP-hard, and the same is true for dominant matching existence (Faenza et al., 2018).

5. Algorithmic paradigms

A central algorithmic tool in one-sided systems is the switching graph. Starting from a popular matching $M$5 in the pruned graph $M$6, one builds a directed graph $M$7 on posts: for each applicant $M$8 and each alternative popular edge $M$9, there is an arc from $M'$0 to $M'$1, weighted by whether $M'$2 prefers the move, dislikes it, or is indifferent. Weight-0 directed cycles and weight-0 paths ending at sinks are switching cycles and switching paths; applying them transforms one popular matching into another. In the presence of ties, every popular matching can be obtained from a fixed popular matching by applying vertex-disjoint switching cycles and switching paths componentwise. This yields an $M'$3-time algorithm to compute all popular pairs when ties are allowed and an $M'$4-time version for strict lists. The same framework also supports single-agent manipulation algorithms that compute optimal cheating strategies with the same asymptotic bounds, and counting popular matchings with ties is $M'$5-complete (Nasre, 2013).

The strict one-sided model also admits efficient parallel algorithms. The first NC algorithm for popular matchings without ties constructs the reduced graph containing only $M'$6- and $M'$7-edges, exploits the fact that each applicant has degree 2 there, and finds an applicant-complete matching via repeated degree-1 eliminations and cycle handling. A second NC algorithm uses the switching graph of a popular matching to compute a maximum-cardinality popular matching. In contrast, for lists with ties, maximum-cardinality bipartite matching is NC-reducible to popular matching, indicating that the parallel complexity of the ties case is at least as difficult as the longstanding NC status of general bipartite matching (Hu et al., 2019).

Popularity also extends beyond one-to-one matchings. In the many-to-many capacitated model $M'$8, where each vertex $M'$9 has capacity $M(v)$0, one compares the sets $M(v)$1 and $M(v)$2 by an adversarial bijection between their symmetric differences and defines a robust vote $M(v)$3 as the minimum over all such pairings. Under this definition, a max-size popular matching can be computed in linear time by the 2-level Gale–Shapley algorithm, which generalizes the classical deferred-acceptance scheme and is justified by an LP-based analysis (Brandl et al., 2016).

6. Robustness, random models, and committee generalizations

Robust popularity asks for a matching that is popular in more than one instance. For pairs of instances that differ only in the preference list of a single agent, robust popular matching is polynomial-time solvable via hybrid instances and the POPULAREDGE primitive. The same approach extends to any fixed number of instances if all changes are confined to the same agent. Hardness emerges quickly beyond that regime: ROBUSTPOPULARMATCHING is NP-complete even when the second instance is obtained by a downward shift of one alternative for exactly four agents. Under altered availability, the problem has a completeness dichotomy: if one instance is complete, robustness under edge deletions is polynomial-time solvable through a maximum-weight popular matching formulation, whereas without completeness the problem is NP-complete even when only two edges are removed (Bullinger et al., 2024).

In random one-sided house allocation, popularity exhibits phase transitions. With complete strict lists and $M(v)$4, Mahdian’s threshold occurs at the unique positive root $M(v)$5 of

$M(v)$6

If $M(v)$7, a popular matching exists with probability $M(v)$8; if $M(v)$9, it exists with probability $M$00. For incomplete strict lists of constant length $M$01, the threshold becomes $M$02, the root of

$M$03

For $M$04, no threshold arises: popular matchings exist with probability $M$05 for all $M$06 (Ruangwises et al., 2016).

When no single popular matching exists, the notion of a popular winning set replaces one matching by a small committee of matchings that collectively do not lose to any single matching. The popular dimension of a problem class is the minimum committee size needed in the worst case. This parameter is exactly 2 for house allocation, even with weighted agents and ties; exactly 1 for the unweighted strict marriage problem; exactly 2 for the unweighted strict roommates problem; and between 2 and 3 for marriage and roommates when weights and/or ties are allowed (Connor et al., 29 Sep 2025). This committee view reframes non-existence not as a failure of majority comparison itself, but as a failure of singleton representation.

Taken together, these lines of work portray popular matchings as a broad family of majority-based equilibrium notions with unusually sharp structural dichotomies. In some regimes—one-sided house allocation, strict one-sided systems, or heavily asymmetric weighted bipartite markets—they admit precise combinatorial characterizations and fast algorithms. In others—weighted two-sided matching, mixed ties, robust multi-instance settings, or non-bipartite existence—they delineate some of the clearest current limits to tractability in algorithmic matching theory (Heeger et al., 2021, Faenza et al., 2018).