Max-Utility Matchings with Popularity via Critical Vertices

We consider the landscape of popular matchings in a bipartite graph G where every vertex has strict preferences over its neighbors. This is a very well-studied model in two-sided matching markets. A matching M is popular if it does not lose a head-to-head election against any matching, where each vertex casts a vote for the matching where it gets a better assignment. Roughly speaking, a popular matching is one such that there is no matching where more vertices are happier. The notion of popularity is more relaxed than stability: a classical notion studied for the last several decades. Popular matchings always exist in G since stable matchings always exist in a bipartite graph and every stable matching is popular. Algorithmically speaking, the landscape of popular matching seems to have only a few bright spots. Every stable matching is a min-size popular matching and there are also simple linear time algorithms for computing a max-size popular matching and for the popular edge problem. All these algorithms reduce the popular matching problem to an appropriate question in stable matchings and solve the corresponding stable matching problem. We now know NP-hardness results for many popular matching problems. These include the min-cost/max-weight popular matching problem and the problem of deciding if G admits a popular matching that is neither a min-size nor a max-size popular matching. For non-bipartite graphs, it is NP-hard to even decide if a popular matching exists or not. A mixed matching is a probability distribution or a lottery over matchings. A popular mixed matching is one that never loses a head-to-head election against any mixed matching. As an allocation mechanism, a popular mixed matching has several nice properties. Moreover, finding a max-weight or min-cost popular mixed matching in G is easy (by solving a linear program). Interestingly, there is always an optimal popular mixed matching Pi with a simple structure: Pi = {(M_0,1/2),(M_1,1/2)} where M_0 and M_1 are matchings in G. Popular mixed matchings always exist in non-bipartite graphs as well and can be computed in polynomial time.

We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if $G$ admits a popular matching that is neither stable nor dominant. We also show a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem. In non-bipartite graphs, we show a strong negative result: it is NP-hard to decide whether a popular matching exists or not, and the same result holds if we replace popular with dominant. On the positive side, we show the following results in any graph: - we identify a subclass of dominant matchings called strongly dominant matchings and show a linear time algorithm to decide if a strongly dominant matching exists or not; - we show an efficient algorithm to compute a popular matching of minimum cost in a graph with edge costs and bounded treewidth.

Let $G = (A \cup B, E)$ be an instance of the stable marriage problem with strict preference lists. A matching $M$ is popular in $G$ if $M$ does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exist and can be easily computed is the set of dominant matchings. A popular matching $M$ is dominant if $M$ wins the head-to-head election against any larger matching. Thus every dominant matching is a max-size popular matching and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that indeed there are differences in the tractability of stable and dominant matchings, and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable, however it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed not only to show positive results on popular matching (as is known), but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp. max-size) popular matching that is not stable (resp. dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when $G$ is non-bipartite.

Our input instance is a bipartite graph G = (A ∪ B, E) where A is a set of applicants, B is a set of jobs, and each vertex u ∊ A ∪ B has a preference list ranking its neighbors in a strict order of preference. For any two matchings M and T in G, let φ (M, T) be the number of vertices that prefer M to T. A matching M is popular if φ(Μ, T) ≥ φ(Τ,M) for all matchings T in G. There is a utility function w : E → ℚ and we consider the problem of matching applicants to jobs in a popular and utility-optimal manner. A popular mixed matching could have a much higher utility than all popular matchings, where a mixed matching is a probability distribution over matchings, i.e., a mixed matching Π = {(M0, p0),…, (Mk, pk)} for some matchings M0,…,Mk and for all i. The function φ(·,) easily extends to mixed matchings; a mixed matching Π is popular if φ(Π,Λ) > φ(Λ, Π) for all mixed matchings Λ in G.Motivated by the fact that a popular mixed matching could have a much higher utility than all popular matchings, we study the popular fractional matching polytope Pg. Our main result is that this polytope is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching Π such that where M0 and M1 are matchings in G. As Π can be computed in polynomial time, an immediate consequence of our result is that in order to implement a max-utility popular mixed matching in G, we need just a single random bit.We analyze PG whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality. In other words, this linear program is identical to its dual program. This is a rare case where an LP of a natural problem has such a property. The self-duality of this LP plays a crucial role in our proof of half-integrality of PG.We also show that our result carries over to the roommates problem, where the graph G need not be bipartite. The polytope of popular fractional matchings is still half-integral here and so we can compute a max-utility popular half-integral matching in G in polynomial time. To complement this result, we also show that the problem of computing a max-utility popular (integral) matching in a roommates instance is NP-hard.

Our input instance is a bipartite graph G = (A ∪ B, E) where A is a set of applicants, B is a set of jobs, and each vertex u ∈ A ∪ B has a preference list ranking its neighbors in a strict order of preference. For any two matchings M and T in G, let ϕ(M,T) be the number of vertices that prefer M to T. A matching M is popular if ϕ(M,T) ≥ ϕ (T,M) for all matchings T in G. There is a utility function w: E → 𝒬 and we consider the problem of matching applicants to jobs in a popular and utility-optimal manner. A popular mixed matching could have a much higher utility than all popular matchings, where a mixed matching is a probability distribution over matchings, i.e., a mixed matching Π = {(Mo, po),..., (Mk,pk)} for some matchings Mo,...,Mk and Σki=0pi = 1, pi ≥ 0 for all i. The function ϕ(·, ·) easily extends to mixed matchings; a mixed matching Π is popular if ϕ (Π, Λ) ≥ ϕ (Λ, Π) for all mixed matchings Λ in G.Motivated by the fact that a popular mixed matching could have a much higher utility than all popular matchings, we study the popular fractional matching polytope PG. Our main result is that this polytope is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching Π such that Π = {(M0, 1/2), (M1, 1/2)} where M0 and M1 are matchings in G. As Π can be computed in polynomial time, an immediate consequence of our result is that in order to implement a max-utility popular mixed matching in G, we need just a single random bit.We analyze PG whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality. In other words, this linear program is identical to its dual program. This is a rare case where an LP of a natural problem has such a property. The self-duality of this LP plays a crucial role in our proof of half-integrality of PG.We also show that our result carries over to the roommates problem, where the graph G need not be bipartite. The polytope of popular fractional matchings is still half-integral here and so we can compute a max-utility popular half-integral matching in G in polynomial time. To complement this result, we also show that the problem of computing a max-utility popular (integral) matching in a roommates instance is NP-hard.

In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time.

Popular matchings in the stable marriage problem

Binary matching in bipartite graphs and its extensions have been well studied over the decades. A stable matching (or marriage) seeks to establish a stable binary pairing of two genders, where each member in a gender has a preference list for the other gender. In addition, all members are paired (also called perfect matching), with one selection made from each gender. Gale and Shapley showed that a stable marriage exists for any set of preferences in any complete and balanced bipartite graphs. However, stable matching may not exist in a non-bipartite graph represented by a complete graph (i.e. single gender), as shown in the stable roommates problem. In this paper, we study binary matching in complete and balanced k-partite graphs with an even number of members, and show that except when k = 2, preference lists always exist under which a stable binary matching does not exist. We use a solution for the stable roommates problem to illustrate the matching identification process if one exists. Under a natural extension of pairwise stability for binary matching to k-ary matching, we show that stable k-ary matching exists for any preference lists in any complete and balanced k-partite graphs. Here, we assume that there is a strict preference order of the members over all individual members from different genders, as opposed to preference order over a combination of members from different genders used in existing multi-dimensional extensions. An extended Gale-Shapley algorithm is introduced, together with various implementations, to determine such a stable k-ary matching. A parallel implementation is also proposed to speed up the matching process. In the extension, we further propose a weakened unstable condition for k-ary matching and show the existence of stable k-ary matching.

Given a bipartite graph G = (A ∪ B, E) where each vertex ranks its neighbors in a strict order of preference, we consider the problem of computing a largest matching in the set of popular matchings in G. A matching M is said to be popular if there is no matching where more vertices are happier than in M. The set of popular matchings is non-empty since every stable matching is popular and it is known that stable matchings always exist in G. The problem of computing a popular matching of maximum size in G = (A ∪ B, E) was considered in [8] where an O(mn0) algorithm was shown, where m = |E| and n0 = min(|A|, |B|). Here we show an O(m) algorithm for this problem. A largest popular matching need not be a maximum cardinality matching in G. There are many applications where maximum cardinality is one of the requirements, and so the next problem that we consider is the following: if M is the set of maximum cardinality matchings in G, compute M* ∊ M that is popular within M. That is, we want a maximum cardinality matching M* such that there is no matching in M that is more popular than M*. We show that such a matching M* always exists and can be computed in O(mn0) time. Though the above matching M* is popular in the set of maximum cardinality matchings, in the entire set of matchings in G, its unpopularity factor could be as high as n0 − 1. On the other hand, a largest popular matching could be of size only ⅔|M*|. In between these two extremes, we show that there is an entire spectrum of matchings: for any integer k, where 2 ≤ k ≤ n0, there is a matching Mk in G of size at least whose unpopularity factor is at most k − 1; also such a matching Mk can be computed in O(mk) time by a simple generalization of our largest popular matching algorithm.

Given a bipartite graph G = (A ∪ B, E) where each vertex ranks its neighbors in a strict order of preference, we consider the problem of computing a largest matching in the set of popular matchings in G. A matching M is said to be popular if there is no matching where more vertices are happier than in M. The set of popular matchings is non-empty since every stable matching is popular and it is known that stable matchings always exist in G. The problem of computing a popular matching of maximum size in G = (A ∪ B, E) was considered in [8] where an O(mn0) algorithm was shown, where m = |E| and n0 = min(|A|, |B|). Here we show an O(m) algorithm for this problem.A largest popular1 matching need not be a maximum cardinality matching in G. There are many applications where maximum cardinality is one of the requirements, and so the next problem that we consider is the following: if M is the set of maximum cardinality matchings in G, compute M* ∈ M that is popular within M. That is, we want a maximum cardinality matching M* such that there is no matching in M that is more popular than M*. We show that such a matching M* always exists and can be computed in O(mn0) time.Though the above matching M* is popular in the set of maximum cardinality matchings, in the entire set of matchings in G, its unpopularity factor could be as high as n0 − 1. On the other hand, a largest popular matching could be of size only 2/3|M*|. In between these two extremes, we show that there is an entire spectrum of matchings: for any integer k, where 2 ≤ k ≤ n0, there is a matching Mk in G of size at least k/k+1|M*| whose unpopularity factor is at most k − 1; also such a matching Mk can be computed in O(mk) time by a simple generalization of our largest popular matching algorithm.

Given a bipartite graph $$G = (A \cup B,E)$$ with strict preference lists and given an edge $$e^* \in E$$ , we ask if there exists a popular matching in G that contains $$e^*$$ . We call this the popular edge problem. A matching M is popular if there is no matching $$M'$$ such that the vertices that prefer $$M'$$ to M outnumber those that prefer M to $$M'$$ . It is known that every stable matching is popular; however G may have no stable matching with the edge $$e^*$$ . In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge $$e^*$$ , then there is either a stable matching that contains $$e^*$$ or a dominant matching that contains $$e^*$$ . This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an $$O(n^3)$$ algorithm to find a popular matching containing a given set of edges or report that none exists, where $$n = |A| + |B|$$ .

The input is a bipartite graph \(G = (\mathcal{A}\cup\mathcal{B}, E)\) where each vertex \(u \in \mathcal{A}\cup\mathcal{B}\) ranks its neighbors in a strict order of preference. A matching M * is said to be popular if there is no matching M such that more vertices are better off in M than in M *. We consider the problem of computing a maximum cardinality popular matching in G. It is known that popular matchings always exist in such an instance G, however the complexity of computing a maximum cardinality popular matching was not known so far. In this paper we give a simple characterization of popular matchings when preference lists are strict and a sufficient condition for a maximum cardinality popular matching. We then show an O(mn 0) algorithm for computing a maximum cardinality popular matching in G, where m = |E| and \(n_0 = \min(|\mathcal{A}|,|\mathcal{B}|)\).KeywordsBipartite GraphStable MatchMaximum CardinalityPreference ListSimple CharacterizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a bipartite graph $$G = A \cup B,E$$ with strict preference lists and given an edge $$e^* \in E$$, we ask if there exists a popular matching in G that containsi¾ź$$e^*$$. We call this the popular edge problem. A matching M is popular if there is no matching $$M'$$ such that the vertices that prefer $$M'$$ to M outnumber those that prefer M toi¾ź$$M'$$. It is known that every stable matching is popular; however G may have no stable matching with the edge $$e^*$$. In this paper we identify another natural subclass of popular called matchings and show that if there is a popular matching that contains the edge $$e^*$$, then there is either a stable matching that contains $$e^*$$ or a dominant matching that containsi¾ź$$e^*$$. This allows us to design a linear time algorithm for the popular edge problem. When preference lists are complete, we show an $$On^3$$ algorithm to find a popular matching containing a given set of edges or report that none exists, where $$n = |A| + |B|$$.

We consider the problem of computing popular matchings in a bipartite graph \(G = (\mathcal {R}\,\cup \, \mathcal {H}, E)\) where \(\mathcal {R}\) and \(\mathcal {H}\) denote a set of residents and a set of hospitals respectively. Each hospital h has a positive capacity denoting the number of residents that can be matched to h. The residents and the hospitals specify strict preferences over each other. This is the well-studied Hospital Residents (HR) problem which is a generalization of the Stable Marriage (SM) problem. The goal is to assign residents to hospitals optimally while respecting the capacities of the hospitals. Stability is a well-accepted notion of optimality in such problems. However, motivated by the need for larger cardinality matchings, alternative notions of optimality like popularity have been investigated in the SM setting. In this paper, we consider a generalized HR setting – namely the Laminar Classified Stable Matchings (LCSM\(^+\)) problem. Here, additionally, hospitals can specify classifications over residents in their preference lists and classes have upper quotas. We show the following new results: We define a notion of popularity and give a structural characterization of popular matchings for the LCSM\(^+\) problem. Assume \(n = |\mathcal {R}| + |\mathcal {H}|\) and \(m = |E|\). We give an O(mn) time algorithm for computing a maximum cardinality popular matching in an LCSM\(^+\) instance. We give an \(O(mn^2)\) time algorithm for computing a matching that is popular amongst the maximum cardinality matchings in an LCSM\(^+\) instance.

We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph G = (R U H, E) where R and H denote sets of residents and hospitals, respectively. Every vertex has a preference list that imposes a strict ordering on its neighbors. In addition, each hospital has an associated upper-quota and a lower-quota. A matching M in G is an assignment of residents to hospitals, and M is said to be feasible if every resident is assigned to at most one hospital and a hospital is assigned at least its lower-quota many residents and at most its upper-quota many residents. Stability is a de-facto notion of optimality in a model where both sets of vertices have preferences. A matching is stable if no unassigned pair has an incentive to deviate from it. It is well-known that an instance of the HRLQ problem need not admit a feasible stable matching. In this paper, we consider the notion of popularity for the HRLQ problem. A matching M is popular if no other matching M' gets more votes than M when vertices vote between M and M'. When there are no lower quotas, there always exists a stable matching and it is known that every stable matching is popular. We show that in an HRLQ instance, although a feasible stable matching need not exist, there is always a matching that is popular in the set of feasible matchings. We give an efficient algorithm to compute a maximum cardinality matching that is popular amongst all the feasible matchings in an HRLQ instance.