Parameterized algorithms for stable matching with ties and incomplete lists

Adil, Deeksha; Gupta, Sushmita; Roy, Sanjukta; Saurabh, Saket; Zehavi, Meirav

doi:10.1016/j.tcs.2018.03.015

Cited by 21 publications

(52 citation statements)

References 13 publications

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“…First, to better understand the NPcompleteness result, one may study the parameterized complexity with respect to the "degree" of incompleteness of the input preferences, such as the number of ties or the number of agents that are in the same equivalence class of the tie-relation. We refer to some recent papers on the parameterized complexity of preference-based stable matching problems [1,11,12,[14][15][16]29,[40][41][42]45] for this line of research. Second, we were not able to settle the computational complexity for complete preferences that are also single-peaked and single-crossing.…”

Section: Resultsmentioning

confidence: 99%

“…contains the rank of agent x in the preference list of agent y. Herein, the rank of agent y in agent x's preference list is defined as the number of agents that are strictly preferred to y by x. For example, if the preference list of agent 1 is 1 2 3 ∼ 4 5 ∼ 6 7, then we may have L 1 =[1,2,3,4,5,6,7] and have R[1,1] = 0, R[1, 2] = 1, R[1, 3] = 2, R[1, 4] = 2, R[1, 5] = 4, R[1, 6] = 4, and R[1, 7] = 6.Before we show the correctness of Algorithm 2, we first claim the following. Each call of InsertMostPreferred(z)(a) assumes D[z] = 0, (b) finds all unmatched agents y which are tied with agent L z [ p z ] and sets A[z, y ] = 1 (Lines (17)-(23)), and (c) pushes a possible matching event {z, y } to Q if, additionally, y also finds z most acceptable (Lines (20)-(21)).…”

mentioning

confidence: 99%

“…The order [U 2c −1,2c ] is defined as follows: For an illustration, assume that E 1 = {e 1 , e 2 } with e 1 = {u 1 , u 2 } and e 2 = {u 5 , u 6 }. Then, [U 1,2 ] may be defined as [U 1,2…”

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confidence: 99%

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Stable roommates with narcissistic, single-peaked, and single-crossing preferences

Bredereck

Finnendahl

Niedermeier

2020

Auton Agent Multi-Agent Syst

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The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommates remains NP-complete.

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Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Stable roommates with narcissistic, single-peaked, and single-crossing preferences

Bredereck

Finnendahl

Niedermeier

2020

Auton Agent Multi-Agent Syst

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“…Proof. Towards a contradiction to (1), suppose that there is a stable matching M with M (a 5 ) / ∈ X. Then, by the preferences of a 5 , there remain three possibilities (i)-(iii) for the partner of a 5 .…”

Section: Incomplete Preferencesmentioning

confidence: 99%

“…Let M be a stable matching for our profile. For the first statement, Lemma 4.2 (1) immediately implies that for every selector agent b 5 j , 1 ≤ j ≤ k, it holds that M (a 5 j ) ∈ {u 10 i |…”

Section: The Above Preferences Are Narcissistic As Well As Single-peamentioning

confidence: 99%

Stable Roommate with Narcissistic, Single-Peaked, and Single-Crossing Preferences

Bredereck

Finnendahl

Niedermeier

2017

Algorithmic Decision Theory

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The classical Stable Roommates problem asks whether it is possible to have a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity-Stable Roommates remains NP-complete. * A preliminary verion of this work appeared in the Proceedings of the 5th International Conference on Algorithmic Decision Theory (ADT '17) [8], volume 10576 of LNCS, pages 315-330, Springer, 2017. This full version contains proof details for Proposition 2.1, Observation 2.2, and Proposition 4.1. Furthermore, the reduction used for our main result (Theorem 4.4) was replaced by a completely new reduction showing NP-hardness for the case of narcissistic, single-peaked, and single-crossing preferences (the preferences in the previous reduction were not narcissistic).

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Algorithms and Complexity of Strongly Stable Non-crossing Matchings

Panda¹,

Sachin²

2023

Lecture Notes in Computer Science

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Parameterized algorithms for stable matching with ties and incomplete lists

Cited by 21 publications

References 13 publications

Stable roommates with narcissistic, single-peaked, and single-crossing preferences

Stable roommates with narcissistic, single-peaked, and single-crossing preferences

Stable Roommate with Narcissistic, Single-Peaked, and Single-Crossing Preferences

Algorithms and Complexity of Strongly Stable Non-crossing Matchings

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