{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T06:58:06Z","timestamp":1771484286117,"version":"3.50.1"},"reference-count":65,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T00:00:00Z","timestamp":1678060800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T00:00:00Z","timestamp":1678060800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["BR 4744\/2-1"],"award-info":[{"award-number":["BR 4744\/2-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P31890"],"award-info":[{"award-number":["P31890"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004063","name":"Knut och Alice Wallenbergs Stiftelse","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004063","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012112","name":"Isaac Newton Institute for Mathematical Sciences","doi-asserted-by":"publisher","award":["EP\/K032208\/1"],"award-info":[{"award-number":["EP\/K032208\/1"]}],"id":[{"id":"10.13039\/100012112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["ERC-StG 639945"],"award-info":[{"award-number":["ERC-StG 639945"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the late 19th century, Swedish mathematician Edvard Phragm\u00e9n proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants\u2014one minimizing the maximum load and one minimizing the variance of loads\u2014and a sequential variant. We study Phragm\u00e9n \u2019s methods from an axiomatic point of view, focusing on properties capturing proportional representation. We show that the sequential variant satisfies <jats:italic>proportional justified representation<\/jats:italic>, which is a rare property for committee monotonic methods. Moreover, we show that the optimization variants satisfy <jats:italic>perfect representation<\/jats:italic>. We also analyze the computational complexity of Phragm\u00e9n \u2019s methods and provide mixed-integer programming based algorithms for computing them.<\/jats:p>","DOI":"10.1007\/s10107-023-01926-8","type":"journal-article","created":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T14:03:48Z","timestamp":1678111428000},"page":"47-76","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Phragm\u00e9n\u2019s voting methods and justified representation"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9509-7017","authenticated-orcid":false,"given":"Markus","family":"Brill","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4744-9449","authenticated-orcid":false,"given":"Rupert","family":"Freeman","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9680-2790","authenticated-orcid":false,"given":"Svante","family":"Janson","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2170-0770","authenticated-orcid":false,"given":"Martin","family":"Lackner","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,6]]},"reference":[{"key":"1926_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00355-019-01208-3","volume":"54","author":"H Aziz","year":"2020","unstructured":"Aziz, H., Lee, B.E.: The expanding approvals rule: improving proportional representation and monotonicity. Soc. Choice Welfare 54, 1\u201345 (2020)","journal-title":"Soc. Choice Welfare"},{"key":"1926_CR2","doi-asserted-by":"crossref","unstructured":"Aziz, H., Lee, B.E.: Proportionally representative participatory budgeting with ordinal preferences. In: Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI), pp. 5110\u20135118. AAAI Press (2021)","DOI":"10.1609\/aaai.v35i6.16646"},{"key":"1926_CR3","doi-asserted-by":"crossref","unstructured":"Aziz, H., Shah, N.: Participatory budgeting: models and approaches. In: Rudas T., P\u00e9li, G., (eds) Pathways Between Social Science and Computational Social Science. Springer, Berlin, pp. 215\u2013236 (2021)","DOI":"10.1007\/978-3-030-54936-7_10"},{"key":"1926_CR4","unstructured":"Aziz, H., Gaspers, S., Gudmundsson, J., Mackenzie, S., Mattei, N., Walsh, T.: Computational aspects of multi-winner approval voting. In: Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pp. 107\u2013115. IFAAMAS (2015)"},{"issue":"2","key":"1926_CR5","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1007\/s00355-016-1019-3","volume":"48","author":"H Aziz","year":"2017","unstructured":"Aziz, H., Brill, M., Conitzer, V., Elkind, E., Freeman, R., Walsh, T.: Justified representation in approval-based committee voting. Soc. Choice Welfare 48(2), 461\u2013485 (2017)","journal-title":"Soc. Choice Welfare"},{"key":"1926_CR6","doi-asserted-by":"crossref","unstructured":"Aziz, H., Elkind, E., Huang, S., Lackner, M., S\u00e1nchez-Fern\u00e1ndez, L., Skowron, P.: On the complexity of extended and proportional justified representation. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence (AAAI), pp. 902\u2013909. AAAI Press (2018a)","DOI":"10.1609\/aaai.v32i1.11478"},{"key":"1926_CR7","unstructured":"Aziz, H., Lee, B.E., Talmon, N.: Proportionally representative participatory budgeting: axioms and algorithms. In: Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pp. 23\u201331. IFAAMAS (2018b)"},{"key":"1926_CR8","unstructured":"Balinski, M., Young, H.P.: Fair representation: meeting the ideal of one man, one vote. Yale University Press, 1982. (2nd Edition [with identical pagination], Brookings Institution Press, 2001)"},{"key":"1926_CR9","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1613\/jair.3896","volume":"47","author":"N Betzler","year":"2013","unstructured":"Betzler, N., Slinko, A., Uhlmann, J.: On the computation of fully proportional representation. J. Artif. Intell. Res. 47, 475\u2013519 (2013)","journal-title":"J. Artif. Intell. Res."},{"key":"1926_CR10","doi-asserted-by":"crossref","unstructured":"Bredereck, R., Faliszewski, P., Kaczmarczyk, A., Niedermeier, R.: An experimental view on committees providing justified representation. In: Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), pp. 109\u2013115. IJCAI (2019)","DOI":"10.24963\/ijcai.2019\/16"},{"key":"1926_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511921698","volume-title":"Modern Computer Arithmetic","author":"RP Brent","year":"2010","unstructured":"Brent, R.P., Zimmermann, P.: Modern Computer Arithmetic, vol. 18. Cambridge University Press, Cambridge (2010)"},{"key":"1926_CR12","doi-asserted-by":"crossref","unstructured":"Brill, M., Freeman, R., Janson, S., Lackner, M.: Phragm\u00e9n\u2019s voting methods and justified representation. In: Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI), pp. 406\u2013413. AAAI Press (2017)","DOI":"10.1609\/aaai.v31i1.10598"},{"issue":"3","key":"1926_CR13","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1177\/0951629818775518","volume":"30","author":"M Brill","year":"2018","unstructured":"Brill, M., Laslier, J.-F., Skowron, P.: Multiwinner approval rules as apportionment methods. J. Theor. Polit. 30(3), 358\u2013382 (2018)","journal-title":"J. Theor. Polit."},{"key":"1926_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01852-1","author":"M Brill","year":"2022","unstructured":"Brill, M., G\u00f6lz, P., Peters, D., Schmidt-Kraepelin, U., Wilker, K.: Approval-based apportionment. Math. Program. (2022). https:\/\/doi.org\/10.1007\/s10107-022-01852-1. (Forthcoming)","journal-title":"Math. Program."},{"key":"1926_CR15","unstructured":"Burdges, J., Cevallos, A., Czaban, P., Habermeier, R., Hosseini, S., Lama, F., Alper, H. K., Luo, X., Shirazi, F., Stewart, A., Wood, G.: Overview of Polkadot and its design considerations. Technical report, arXiv:2005.13456 [cs.CR] (2020)"},{"issue":"9","key":"1926_CR16","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1080\/00029890.1924.11986375","volume":"31","author":"WD Cairns","year":"1924","unstructured":"Cairns, W.D.: The international mathematical congress at Toronto. Am. Math. Mon. 31(9), 411\u2013417 (1924)","journal-title":"Am. Math. Mon."},{"key":"1926_CR17","unstructured":"Camps, R., Mora, X., Saumell, L.: The method of Enestr\u00f6m and Phragm\u00e9n for parliamentary elections by means of approval voting. Technical report, arXiv:1907.10590 [econ.TH] (2019)"},{"key":"1926_CR18","doi-asserted-by":"crossref","unstructured":"Cevallos, A., Stewart, A.: A verifiably secure and proportional committee election rule. In: Proceedings of the 3rd ACM Conference on Advances in Financial Technologies (AFT), pp. 29\u201342. ACM (2021)","DOI":"10.1145\/3479722.3480988"},{"issue":"3","key":"1926_CR19","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1007\/s00355-017-1026-z","volume":"48","author":"E Elkind","year":"2017","unstructured":"Elkind, E., Faliszewski, P., Skowron, P., Slinko, A.: Properties of multiwinner voting rules. Soc. Choice Welfare 48(3), 599\u2013632 (2017)","journal-title":"Soc. Choice Welfare"},{"key":"1926_CR20","unstructured":"Elkind, E., Lackner, M., Peters, D.: Structured preferences. In: Endriss, U., (ed), Trends in Computational Social Choice, chapter 10, pp. 187\u2013207. AI Access (2017)"},{"key":"1926_CR21","unstructured":"Faliszewski, P., Skowron, P., Slinko, A., Talmon, N.: Multiwinner voting: a new challenge for social choice theory. In: Endriss, U., (ed), Trends in Computational Social Choice, chapter 2. AI Access (2017)"},{"key":"1926_CR22","unstructured":"Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)"},{"issue":"3","key":"1926_CR23","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/0304-3975(76)90059-1","volume":"1","author":"MR Garey","year":"1976","unstructured":"Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237\u2013267 (1976)","journal-title":"Theor. Comput. Sci."},{"key":"1926_CR24","doi-asserted-by":"crossref","unstructured":"Israel, J., Brill, M.: Dynamic proportional rankings. In: Proceedings of the 30th International Joint Conference on Artificial Intelligence (IJCAI), pp. 261\u2013267. IJCAI (2021)","DOI":"10.24963\/ijcai.2021\/37"},{"key":"1926_CR25","unstructured":"Janson, S.: Proportionella valmetoder. Unpublished manuscript. Available at http:\/\/www2.math.uu.se\/~svante\/papers\/sjV6.pdf (2012)"},{"key":"1926_CR26","unstructured":"Janson, S.: Phragm\u00e9n\u2019s and Thiele\u2019s election methods. Technical report, arXiv:1611.08826v2 [math.HO] (2018)"},{"key":"1926_CR27","unstructured":"Janson, S.: Thresholds quantifying proportionality criteria for election methods. Technical report, arXiv:1810.06377 [cs.GT] (2018)"},{"key":"1926_CR28","doi-asserted-by":"crossref","unstructured":"Jaworski, M., Skowron, P.: Phragm\u00e9n rules for degressive and regressive proportionality. In: Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI), pp. 328\u2013334. IJCAI (2022)","DOI":"10.24963\/ijcai.2022\/47"},{"key":"1926_CR29","doi-asserted-by":"crossref","unstructured":"Kilgour, D.M.: Approval balloting for multi-winner elections. In: Handbook on Approval Voting, chapter 6. Springer (2010)","DOI":"10.1007\/978-3-642-02839-7_6"},{"key":"1926_CR30","doi-asserted-by":"crossref","unstructured":"Lackner, M., Maly, J.: Proportional decisions in perpetual voting. In: Proceedings of the 37th AAAI Conference on Artificial Intelligence (AAAI). AAAI Press (2023). (Forthcoming)","DOI":"10.1609\/aaai.v37i5.25710"},{"key":"1926_CR31","volume-title":"Multi-Winner Voting with Approval Preferences","author":"M Lackner","year":"2022","unstructured":"Lackner, M., Skowron, P.: Multi-Winner Voting with Approval Preferences. Springer, Berlin (2022)"},{"key":"1926_CR32","unstructured":"Lu, T., Boutilier, C.: Budgeted social choice: from consensus to personalized decision making. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), pp. 280\u2013286. AAAI Press (2011)"},{"issue":"261","key":"1926_CR33","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1090\/S0025-5718-07-02017-0","volume":"77","author":"N M\u00f6ller","year":"2008","unstructured":"M\u00f6ller, N.: On Sch\u00f6nhage\u2019s algorithm and subquadratic integer GCD computation. Math. Comput. 77(261), 589\u2013607 (2008)","journal-title":"Math. Comput."},{"issue":"4","key":"1926_CR34","doi-asserted-by":"publisher","first-page":"925","DOI":"10.2307\/2082518","volume":"89","author":"BL Monroe","year":"1995","unstructured":"Monroe, B.L.: Fully proportional representation. Am. Polit. Sci. Rev. 89(4), 925\u2013940 (1995)","journal-title":"Am. Polit. Sci. Rev."},{"key":"1926_CR35","unstructured":"Mora, X.: Phragm\u00e9n\u2019s sequential method with a variance criterion. Technical report, arXiv:1611.06833 [math.OC] (2016)"},{"issue":"1","key":"1926_CR36","first-page":"57","volume":"30","author":"X Mora","year":"2015","unstructured":"Mora, X., Oliver, M.: Eleccions mitjan\u00e7ant el vot d\u2019aprovaci\u00f3. El m\u00e8tode de Phragm\u00e9n i algunes variants. Butl. Soc. Catalana Mat. 30(1), 57\u2013101 (2015)","journal-title":"Butl. Soc. Catalana Mat."},{"key":"1926_CR37","doi-asserted-by":"publisher","DOI":"10.1017\/CCOL0521360552","volume-title":"Axioms of Cooperative Decision Making","author":"H Moulin","year":"1988","unstructured":"Moulin, H.: Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge (1988)"},{"issue":"3","key":"1926_CR38","doi-asserted-by":"publisher","first-page":"566","DOI":"10.1016\/S0377-2217(96)00154-3","volume":"100","author":"W Ogryczak","year":"1997","unstructured":"Ogryczak, W.: On the lexicographic minimax approach to location problems. Eur. J. Oper. Res. 100(3), 566\u2013585 (1997)","journal-title":"Eur. J. Oper. Res."},{"key":"1926_CR39","doi-asserted-by":"crossref","unstructured":"Ogryczak, W., Sliwinski, T.: On direct methods for lexicographic min-max optimization. In: Gavrilova, M.L., Gervasi, O., Kumar, V., Tan, C.J.K., Taniar, D., Lagan\u00e0, A., Mun, Y., Choo, H. (eds) Computational Science and Its Applications - ICCSA 2006, volume 3982 of Lecture Notes in Computer Science, pp. 802\u2013811. Springer (2006)","DOI":"10.1007\/11751595_85"},{"key":"1926_CR40","doi-asserted-by":"crossref","unstructured":"Peters, D., Skowron, P.: Proportionality and the limits of welfarism. In: Proceedings of the 21st ACM Conference on Economics and Computation (ACM-EC), pp. 793\u2013794 (2020). Full version arXiv:1911.11747 [cs.GT]","DOI":"10.1145\/3391403.3399465"},{"key":"1926_CR41","unstructured":"Peters, D., Pierczy\u0144ski, G., Skowron, P.: Proportional participatory budgeting with additive utilities. In: Proceedings of the 35th Annual Conference on Neural Information Processing Systems (NeurIPS), pp. 12726\u201312737 (2021)"},{"key":"1926_CR42","unstructured":"Phragm\u00e9n, E.: Om proportionella val. Stockholms Dagblad, 14 March 1893 (1893). Summary of a public lecture published in a newspaper"},{"issue":"3","key":"1926_CR43","first-page":"133","volume":"51","author":"E Phragm\u00e9n","year":"1894","unstructured":"Phragm\u00e9n, E.: Sur une m\u00e9thode nouvelle pour r\u00e9aliser, dans les \u00e9lections, la repr\u00e9sentation proportionnelle des partis. \u00d6fversigt af Kongliga Vetenskaps-Akademiens F\u00f6rhandlingar 51(3), 133\u2013137 (1894)","journal-title":"\u00d6fversigt af Kongliga Vetenskaps-Akademiens F\u00f6rhandlingar"},{"key":"1926_CR44","unstructured":"Phragm\u00e9n, E.: Proportionella val. En valteknisk studie. Svenska sp\u00f6rsm\u00e5l 25. Lars H\u00f6kersbergs f\u00f6rlag, Stockholm (1895)"},{"key":"1926_CR45","first-page":"181","volume":"53","author":"E Phragm\u00e9n","year":"1896","unstructured":"Phragm\u00e9n, E.: Sur la th\u00e9orie des \u00e9lections multiples. \u00d6fversigt af Kongliga Vetenskaps-Akademiens F\u00f6rhandlingar 53, 181\u2013191 (1896)","journal-title":"\u00d6fversigt af Kongliga Vetenskaps-Akademiens F\u00f6rhandlingar"},{"issue":"2","key":"1926_CR46","first-page":"297","volume":"2","author":"E Phragm\u00e9n","year":"1899","unstructured":"Phragm\u00e9n, E.: Till fr\u00e5gan om en proportionell valmetod. Statsvetenskaplig Tidskrift 2(2), 297\u2013305 (1899)","journal-title":"Statsvetenskaplig Tidskrift"},{"issue":"1","key":"1926_CR47","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/BF02415450","volume":"31","author":"E Phragm\u00e9n","year":"1908","unstructured":"Phragm\u00e9n, E., Lindel\u00f6f, E.: Sur une extension d\u2019un principe classique de l\u2019analyse et sur quelques propri\u00e9t\u00e9s des fonctions monog\u00e8nes dans le voisinage d\u2019un point singulier. Acta Math. 31(1), 381\u2013406 (1908)","journal-title":"Acta Math."},{"key":"1926_CR48","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s10107-016-1036-0","volume":"162","author":"AD Pia","year":"2017","unstructured":"Pia, A.D., Dey, S.S., Molinaro, M.: Mixed-integer quadratic programming is in NP. Math. Program. 162, 225\u2013240 (2017)","journal-title":"Math. Program."},{"key":"1926_CR49","unstructured":"Polkadot Wiki. NPoS election algorithms. https:\/\/wiki.polkadot.network\/docs\/learn-phragmen (2021). Accessed: 01 Jan 2023"},{"issue":"2","key":"1926_CR50","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1177\/0951692898010002001","volume":"10","author":"RF Potthof","year":"1998","unstructured":"Potthof, R.F., Brams, S.J.: Proportional representation: broadening the options. J. Theor. Polit. 10(2), 147\u2013178 (1998)","journal-title":"J. Theor. Polit."},{"key":"1926_CR51","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/s00355-007-0235-2","volume":"30","author":"AD Procaccia","year":"2008","unstructured":"Procaccia, A.D., Rosenschein, J.S., Zohar, A.: On the complexity of achieving proportional representation. Soc. Choice Welfare 30, 353\u2013362 (2008)","journal-title":"Soc. Choice Welfare"},{"key":"1926_CR52","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-03856-8","volume-title":"Proportional Representation: Apportionment Methods and Their Applications","author":"F Pukelsheim","year":"2014","unstructured":"Pukelsheim, F.: Proportional Representation: Apportionment Methods and Their Applications. Springer, Berlin (2014)"},{"key":"1926_CR53","doi-asserted-by":"publisher","DOI":"10.1177\/13540688211066711","author":"A Rosenfeld","year":"2022","unstructured":"Rosenfeld, A., Shapiro, E., Talmon, N.: Proportional ranking in primary elections: a case study. Party Politics (2022). https:\/\/doi.org\/10.1177\/13540688211066711. (Forthcoming)","journal-title":"Party Politics"},{"key":"1926_CR54","unstructured":"S\u00e1nchez-Fern\u00e1ndez, L., Fisteus, J.A.: Monotonicity axioms in approval-based multi-winner voting rules. In: Proceedings of the 18th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pp. 485\u2013493. IFAAMAS (2019). Full version arXiv:1710.04246v3 [cs.GT]"},{"key":"1926_CR55","unstructured":"S\u00e1nchez-Fern\u00e1ndez, L., Elkind, E., Lackner, M.: Committees providing EJR can be computed efficiently. Technical report, arXiv:1704.00356v3 [cs.GT] (2017a)"},{"key":"1926_CR56","doi-asserted-by":"crossref","unstructured":"S\u00e1nchez-Fern\u00e1ndez, L., Elkind, E., Lackner, M., Fern\u00e1ndez, N., Fisteus, J.A., Basanta Val, P., Skowron, P.: Proportional justified representation. In: Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI), pp. 670\u2013676. AAAI Press (2017b)","DOI":"10.1609\/aaai.v31i1.10611"},{"key":"1926_CR57","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01805-8","author":"L S\u00e1nchez-Fern\u00e1ndez","year":"2022","unstructured":"S\u00e1nchez-Fern\u00e1ndez, L., Fern\u00e1ndez, N., Fisteus, J.A., Brill, M.: The maximin support method: an extension of the D\u2019Hondt method to approval-based multiwinner elections. Math. Program. (2022). https:\/\/doi.org\/10.1007\/s10107-022-01805-8. (Forthcoming)","journal-title":"Math. Program."},{"issue":"6","key":"1926_CR58","doi-asserted-by":"publisher","first-page":"1163","DOI":"10.1137\/0117107","volume":"17","author":"D Schmeidler","year":"1969","unstructured":"Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17(6), 1163\u20131170 (1969)","journal-title":"SIAM J. Appl. Math."},{"key":"1926_CR59","volume-title":"Theory of Linear and Integer Programming","author":"A Schrijver","year":"1986","unstructured":"Schrijver, A.: Theory of Linear and Integer Programming. Wiley, London (1986)"},{"key":"1926_CR60","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1016\/j.artint.2016.09.003","volume":"241","author":"P Skowron","year":"2016","unstructured":"Skowron, P., Faliszewski, P., Lang, J.: Finding a collective set of items: from proportional multirepresentation to group recommendation. Artif. Intell. 241, 191\u2013216 (2016)","journal-title":"Artif. Intell."},{"key":"1926_CR61","doi-asserted-by":"crossref","unstructured":"Skowron, P., Lackner, M., Brill, M., Peters, D., Elkind, E.: Proportional rankings. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), pp. 409\u2013415. IJCAI (2017)","DOI":"10.24963\/ijcai.2017\/58"},{"issue":"3\u20134","key":"1926_CR62","first-page":"281","volume":"7","author":"ASV Strassen","year":"1971","unstructured":"Strassen, A.S.V.: Schnelle Multiplikation gro\u00dfer Zahlen. Computing 7(3\u20134), 281\u2013292 (1971)","journal-title":"Computing"},{"key":"1926_CR63","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-11672-8","volume-title":"G\u00f6sta Mittag-Leffler: A man of conviction","author":"A Stubhaug","year":"2010","unstructured":"Stubhaug, A.: G\u00f6sta Mittag-Leffler: A man of conviction. Springer, Berlin (2010)"},{"key":"1926_CR64","unstructured":"Thiele, T.N.: Om flerfoldsvalg. Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger, pp. 415\u2013441 (1895)"},{"issue":"1","key":"1926_CR65","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1257\/jep.9.1.27","volume":"9","author":"N Tideman","year":"1995","unstructured":"Tideman, N.: The single transferable vote. J. Econ. Perspect. 9(1), 27\u201338 (1995)","journal-title":"J. Econ. Perspect."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-023-01926-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-023-01926-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-023-01926-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T18:09:51Z","timestamp":1707502191000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-023-01926-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,6]]},"references-count":65,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["1926"],"URL":"https:\/\/doi.org\/10.1007\/s10107-023-01926-8","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,6]]},"assertion":[{"value":"31 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors declare that they have no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}]}}