Popular Matchings in Complete Graphs

Cseh, Ágnes; Kavitha, Telikepalli

doi:10.1007/s00453-020-00791-7

Cited by 9 publications

(5 citation statements)

References 31 publications

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“…For graphs with an even n, our algorithm is only able to decide whether a non-perfect popular matching exists. This is not surprising, because deciding whether a (perfect) popular matching exists in a complete graph with an even n is NP-complete [7], while our algorithm runs in polynomial time if |U | is small and the minimum degree in the graph is large.…”

Section: Correctness and Running Timementioning

confidence: 86%

“…Only recently Faenza et al [12] and Gupta et al [15] resolved the long-standing [2,5,18,20,29] open question on the complexity of deciding whether a popular matching exists in a popular roommates instance and showed that the problem is NP-complete. This hardness result remains valid even if the preference lists are complete [7]. However, this hardness result is only valid for complete graphs with an even number of vertices.…”

Section: Introductionmentioning

confidence: 91%

“…Decomposing a popular matching to a set of edges that is stable on their subgraph and to a set of edges that is popular on their subgraph appears in two further papers. Cseh and Kavitha [7] identified the set of edges that can appear in a popular matching on a bipartite instance by showing that any popular matching M can be decomposed as M = M 0 ∪ M 1 , where M 0 is a so-called dominant matching in the subgraph induced by the vertices matched in M 0 , and in the subgraph induced by the remaining vertices, M 1 is stable. Kavitha [25] searched for popular matchings in non-bipartite instances by showing that every popular matching can be partitioned into a stable part and a truly popular part.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

“…However, this hardness result is only valid for complete graphs with an even number of vertices. It is known that when n, the number of vertices in G is odd, a matching in a complete graph G on n vertices is popular only if it is stable [7]. Since Irving's algorithm can decide if G admits a stable matching or not [22], the popular roommates problem in a complete graph G can be efficiently solved for odd n.…”

Section: Introductionmentioning

confidence: 99%

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Polynomially tractable cases in the popular roommates problem

Bérczi-Kovács,

Cseh,

Kosztolányi

et al. 2021

Preprint

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The input of the popular roommates problem consists of a graph G = (V, E) and for each vertex v ∈ V , strict preferences over the neighbors of v. Matching M is more popular than M ′ if the number of vertices preferring M to M ′ is larger than the number of vertices preferring M ′ to M . A matching M is called popular if there is no matching M ′ that is more popular than M .Only recently Faenza et al. [12] and Gupta et al.[15] resolved the long-standing open question on the complexity of deciding whether a popular matching exists in a popular roommates instance and showed that the problem is NP-complete. In this paper we identify a class of instances that admit a polynomial-time algorithm for the problem. We also test these theoretical findings on randomly generated instances to determine the existence probability of a popular matching in them.

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Section: Correctness and Running Timementioning

confidence: 86%

Section: Introductionmentioning

confidence: 91%

Section: Conclusion and Open Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Polynomially tractable cases in the popular roommates problem

Bérczi-Kovács,

Cseh,

Kosztolányi

et al. 2021

Preprint

Self Cite

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show abstract

“…Recently, Faenza et al [6] and Gupta et al [10] independently proved that this problem is still NPhard for rp even when people's preference lists are strict. Cseh and Kavitha [4] showed that, in a complete graph rp instance where each person's preference list is strict and contains all other people, the problem of determining whether a popular matching exists can be solved in polynomial time for an odd n but is NP-hard for an even n.…”

Section: Popular Matchingsmentioning

confidence: 99%

Unpopularity Factor in the Marriage and Roommates Problems

Ruangwises

Itoh

2020

Theory Comput Syst

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Given a set A of n people, with each person having a preference list that ranks a subset of A as his/her acceptable partners in order of preference, we consider the Roommates Problem (rp) and the Marriage Problem (mp) of matching people with their partners. In rp there is no further restriction, while in mp only people with opposite genders can be matched. For a pair of matchings X and Y , let φ(X, Y ) denote the number of people who prefer the person they get matched by X to the person they get matched by Y , and define an unpopularity factor u(M ) of a matching M to be the maximum ratio φ(M ′ , M )/φ(M, M ′ ) among all other possible matchings M ′ . In this paper, we develop an algorithm to compute the unpopularity factor of a given matching in O(m √ n log 2 n) time for rp and in O(m √ n log n) time for mp, where m is the total length of people's preference lists. We also generalize the notion of unpopularity factor to the weighted setting where people are given different voting weights and show that our algorithm can be slightly modified to support that setting with the same running time.

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