Popular Matchings: Structure and Algorithms

McDermid, Eric; Irving, Robert W.

doi:10.1007/978-3-642-02882-3_50

Cited by 22 publications

(31 citation statements)

References 9 publications

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“…We now give a switching graph characterization of popular matchings for instances from this class. Our results are motivated from similar characterizations for HA in [12] and for HAT in [13]. A switching graph for an instance allows us to move from one popular matching to another by making well defined walks on the switching graph.…”

Section: Switching Graph Characterization Of Chamentioning

confidence: 93%

“…Our motivation for this study is largely due to the similarity of structures between stable matchings and popular matchings (although no direct relationship is known). The interest is further fueled by the existence of a linear time algorithm to exactly count the number of popular matchings in the standard setting [12]. We look at generalizations of the standard version -preferences with ties and houses with capacities.…”

Section: Introductionmentioning

confidence: 99%

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Counting Popular Matchings in House Allocation Problems

Acharyya

Chakraborty

Jha

2014

Computer Science - Theory and Applications

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We study the problem of counting the number of popular matchings in a given instance. A popular matching instance consists of agents A and houses H, where each agent ranks a subset of houses according to their preferences. A matching is an assignment of agents to houses. A matching M is more popular than matching M ′ if the number of agents that prefer M to M ′ is more than the number of people that prefer M ′ to M . A matching M is called popular if there exists no matching more popular than M . McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered.We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem.We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard.

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Section: Switching Graph Characterization Of Chamentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Counting Popular Matchings in House Allocation Problems

Acharyya

Chakraborty

Jha

2014

Computer Science - Theory and Applications

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“…To choose one rank-maximal matching, it is natural to impose an additional optimality criterion. Such a question has been considered earlier in the context of popular matchings by [9,11] and also in the context of the stable marriage problem [7]. The additional notion of optimality that we impose is the notion of popularity, defined below:…”

Section: Popularity Of Rank-maximal Matchingsmentioning

confidence: 99%

Rank-Maximal Matchings – Structure and Algorithms

Ghosal

Nasre

Nimbhorkar

2014

Algorithms and Computation

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Let G = (A ∪ P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rankmaximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rankmaximal matchings is #P -Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rankmaximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem.⋆ Part of the work has been done while the author was on a sabbatical to the Institute of Mathematics of the Czech Academy of Sciences, Prague.

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“…The generalizations include the capacitated version studied by Manlove and Sng [9], the weighted version studied by Mestre [12] and random popular matchings studied by Mahdian [8]. Kavitha and Nasre [6] as well as McDermind and Irving [11] independently studied the problem of computing an optimal popular matching for strict instances where the notion of optimality is specified as a part of the input. Note that they also considered the min-cost popular matchings but in this version the costs are associated with edges whereas in our problem costs are associated with items.…”

Section: Introductionmentioning

confidence: 99%

Popularity at minimum cost

2012

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Abstract. We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph G = (A ∪ B, E), where A is a set of people, B is a set of items, and each person a ∈ A ranks a subset of items in an order of preference, with ties allowed. The popular matching problem seeks to compute a matching M * between people and items such that there is no matching M where more people are happier with M than with M * . Such a matching M * is called a popular matching. However, there are simple instances where no popular matching exists.Here we consider the following natural extension to the above problem: associated with each item b ∈ B is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to "augment" G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of √ n1/2, where n1 is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn1) time, where m is the number of edges.

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Popular Matchings: Structure and Algorithms

Cited by 22 publications

References 9 publications

Counting Popular Matchings in House Allocation Problems

Counting Popular Matchings in House Allocation Problems

Rank-Maximal Matchings – Structure and Algorithms

Popularity at minimum cost

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