Characterization of stable matchings as extreme points of a polytope

Rothblum, Uriel G.

doi:10.1007/bf01586041

Cited by 118 publications

(111 citation statements)

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“…A stable matching satisfying all constraints on restricted edges exists if and only if there is a stable matching of weight −|Q| in the weighted instance, where Q is the set of forced edges. With the help of rotations, minimum weight stable matchings can be found in polynomial time [11][12][13][14] (see the final paragraph of Section 2 for more details on the role played by each of these references).…”

Section: Theorem 12 (Dias Et Al [10]) the Problem Of Finding A Stamentioning

confidence: 99%

“…Redesigning the weight function to avoid the monotonicity requirement using Feder's method can radically increase K. Fortunately, linear programming techniques allow the conditions to be dropped while retaining polynomial-time solvability. A simple and elegant formulation of the sm polytope is known [13] and using this, a minimum weight stable matching can be computed for all real-valued weight functions in polynomial time via linear programming. For weighted sr, finding an optimal matching is NP-hard, but 2-approximable with combinatorial methods, under the assumption of monotone, non-negative and integral weights [12].…”

Section: Preliminaries and Techniquesmentioning

confidence: 99%

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Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

Cseh

Manlove

2015

Algorithmic Game Theory

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a b s t r a c tIn the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to NP-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an NP-hard but (under some cardinality assumptions) 2-approximable problem. In the case of NP-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs.

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Section: Theorem 12 (Dias Et Al [10]) the Problem Of Finding A Stamentioning

confidence: 99%

Section: Preliminaries and Techniquesmentioning

confidence: 99%

Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

Cseh

Manlove

2015

Algorithmic Game Theory

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“…Another, quite different approach to providing parallel proofs of the common results for the marriage and assignment models derives from the demonstration in Vande Vate (1989) that the set of sta ble matchings in the marriage model can be formulated, as in the assignment model, as the solutions to a linear programming prob lem (see also Rothblum, 1991). Althoug' h the linear programming formulations for these two problems are quite different from one an other, Roth, Rothblum, and Vande Vate (1991) showed that they could be used to provide similar proofs of some of the common re sults, using the duality theorem of linear programming.…”

Section: And Andmentioning

confidence: 99%

Stable Outcomes in Discrete and Continuous Models of Two-Sided Matching: a Unified Treatment

Roth

Sotomayor

1996

Brazilian Review of Econometrics

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We present a unified treatment of a class of two-sided matching models that includes discrete models (such as the marriage model of Gale and Shapley) and continuous models (such as the assignmnent model of Shapley aud Shubik and the generalized assignment model of Demange aud Gale). In contrast with previous, treatments, the parallel conclusions for the two sets of models are derived here in the same way from the same assumptions. We show that the results ill question all follow closely from the assumptions that the core coincides with the core defined by weak domination. In the marriage model, the assumption of strict preferences causes these two sets to coincide, while in the continuous models the two sets coincide because agents have continuous preferences and prices can be adjusted continuously.

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“…There are many polynomial time algorithms to compute a max-utility stable matching in G [12,13,15,22,33,36,37] -several of these use linear programming on the stable matching polytope S G , which is the convex hull of the 0-1 edge incidence vectors of stable matchings in G. The utility of a max-utility popular matching could be much more than that of a max-utility stable matching; for instance, when all utilities are 1, a max-utility popular matching is the same as a max-size popular matching and a stable matching is a minsize popular matching [20].…”

Section: Introductionmentioning

confidence: 99%

“…Roth [31,32] discusses how stable matchings compare in practice with other types of matchings in the two-sided matching markets. The first description of the polytope S G for G = (A ∪ B, E) was given by Vande Vate [37] in 1989 and several descriptions of S G are now known [11,15,30,33,36].…”

Section: Introductionmentioning

confidence: 99%

Popularity, Mixed Matchings, and Self-duality

Huang¹,

Kavitha²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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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 preferThere 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 matchingMotivated 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 P G . 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 matching2 )} where M 0 and M 1 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 P G 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 P G .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.

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Characterization of stable matchings as extreme points of a polytope

Cited by 118 publications

References 4 publications

Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

Stable Outcomes in Discrete and Continuous Models of Two-Sided Matching: a Unified Treatment

Popularity, Mixed Matchings, and Self-duality

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