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Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathsf {NP}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>NP<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440\u2013465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to\u00a01, which generalizes the setting of \u201carranged marriages\u201d first considered by Knuth (Mariages stables et leurs relations avec d\u2019autres probl\u00e8mes combinatoires, Les Presses de l\u2019Universit\u00e9 de Montr\u00e9al, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomial-time algorithm, or an <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathsf {NP}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>NP<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hardness proof and fixed-parameter algorithm, or <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathsf {NP}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>NP<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hardness proof and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathsf {W}[1]$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>W<\/mml:mi><mml:mo>[<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440\u2013465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.<\/jats:p>","DOI":"10.1007\/s00453-019-00636-y","type":"journal-article","created":{"date-parts":[[2020,2,6]],"date-time":"2020-02-06T14:03:22Z","timestamp":1580997802000},"page":"1136-1188","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Stable Matchings with Covering Constraints: A Complete Computational Trichotomy"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4721-5354","authenticated-orcid":false,"given":"Matthias","family":"Mnich","sequence":"first","affiliation":[]},{"given":"Ildik\u00f3","family":"Schlotter","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,6]]},"reference":[{"issue":"1","key":"636_CR1","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/s00453-016-0252-6","volume":"8","author":"A Arulselvan","year":"2018","unstructured":"Arulselvan, A., Cseh, \u00c1., Gro\u00df, M., Manlove, D.F., Matuschke, J.: Matchings with lower quotas: algorithms and complexity. 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