{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T14:27:31Z","timestamp":1787495251535,"version":"3.56.0"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2012,5,25]],"date-time":"2012-05-25T00:00:00Z","timestamp":1337904000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Event Dyn Syst"],"published-print":{"date-parts":[[2013,6]]},"DOI":"10.1007\/s10626-012-0137-z","type":"journal-article","created":{"date-parts":[[2012,8,30]],"date-time":"2012-08-30T10:17:05Z","timestamp":1346321825000},"page":"105-134","source":"Crossref","is-referenced-by-count":12,"title":["The level set method for the two-sided max-plus eigenproblem"],"prefix":"10.1007","volume":"23","author":[{"given":"St\u00e9phane","family":"Gaubert","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Serge\u012d","family":"Sergeev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2012,5,25]]},"reference":[{"key":"137_CR1","first-page":"927","volume":"327","author":"M Akian","year":"1998","unstructured":"Akian M, Bapat R, Gaubert S (1998) Asymptotics of the Perron eigenvalue and eigenvector using max-algebra. Comptes Rendus Acad Sci Ser I 327:927\u2013932","journal-title":"Comptes Rendus Acad Sci Ser I"},{"key":"137_CR2","first-page":"103","volume":"339","author":"M Akian","year":"2004","unstructured":"Akian M, Bapat R, Gaubert S (2004) Perturbation of eigenvalues of matrix pencils and optimal assignment problem. Comptes Rendus Acad Sci Ser I 339:103\u2013108. arXiv:math\/0402438","journal-title":"Comptes Rendus Acad Sci Ser I"},{"key":"137_CR3","unstructured":"Akian M, Bapat R, Gaubert S (2004\u20132006) Min-plus methods in eigenvalue perturbation theory and generalized Lidski\u012d\u2013Vi\u0161ik\u2013Ljusternik theorem. arXiv:math\/0402090v3"},{"key":"137_CR4","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1090\/conm\/377\/06983","volume-title":"Idempotent mathematics and mathematical physics. Cont Math, vol 377","author":"M Akian","year":"2005","unstructured":"Akian M, Gaubert S, Kolokoltsov V (2005) Set coverings and invertibility of the functional galois connections. In: Litvinov GL, Maslov VP (eds) Idempotent mathematics and mathematical physics. Cont Math, vol 377. AMS, Providence, pp 19\u201351. arXiv:math.FA\/0403441"},{"key":"137_CR5","doi-asserted-by":"crossref","unstructured":"Akian M, Bapat R, Gaubert S (2006) Max-plus algebras. In: Hogben L (ed) Handbook of linear algebra (discrete mathematics and its applications), vol\u00a039, chap\u00a025. Chapman & Hall\/CRC","DOI":"10.1201\/9781420010572.ch25"},{"issue":"12","key":"137_CR6","doi-asserted-by":"crossref","first-page":"3261","DOI":"10.1016\/j.laa.2011.06.009","volume":"435","author":"M Akian","year":"2011","unstructured":"Akian M, Gaubert S, Nitica V, Singer I (2011) Best approximation in max-plus semimodules. Linear Algebra Appl 435(12):3261\u20133296. arXiv:1012.5492","journal-title":"Linear Algebra Appl"},{"issue":"1","key":"137_CR7","doi-asserted-by":"crossref","first-page":"125001","DOI":"10.1142\/S0218196711006674","volume":"22","author":"M Akian","year":"2012","unstructured":"Akian M, Gaubert S, Guterman A (2012) Tropical polyhedra are equivalent to mean payoff games. Int J Algebra Comput 22(1):125001 (43 pages). doi: 10.1142\/S0218196711006674 , arXiv:0912.2462","journal-title":"Int J Algebra Comput"},{"issue":"7","key":"137_CR8","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1016\/j.laa.2011.02.004","volume":"435","author":"X Allamigeon","year":"2011","unstructured":"Allamigeon X, Gaubert S, Katz RD (2011) Tropical polar cones, hypergraph transversals, and mean payoff games. Linear Algebra Appl 435(7):1549\u20131574. arXiv:1004.2778","journal-title":"Linear Algebra Appl"},{"key":"137_CR9","unstructured":"Baccelli FL, Cohen G, Olsder GJ, Quadrat JP (1992) Synchronization and linearity: an algebra for discrete event systems. Wiley"},{"key":"137_CR10","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.laa.2006.09.023","volume":"422","author":"P Binding","year":"2007","unstructured":"Binding P, Volkmer H (2007) A generalized eigenvalue problem in the max algebra. Linear Algebra Appl 422:360\u2013371","journal-title":"Linear Algebra Appl"},{"key":"137_CR11","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.dam.2006.04.029","volume":"155","author":"H Bjorklund","year":"2007","unstructured":"Bjorklund H, Vorobyov S (2007) A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games. Discrete Appl Math 155:210\u2013229","journal-title":"Discrete Appl Math"},{"key":"137_CR12","doi-asserted-by":"crossref","unstructured":"Burns SM (1991) Performance analysis and optimization of asynchronous circuits. PhD thesis, California Institute of Technology","DOI":"10.21236\/ADA447732"},{"issue":"11","key":"137_CR13","doi-asserted-by":"crossref","first-page":"1306","DOI":"10.1109\/12.475126","volume":"44","author":"SM Burns","year":"1995","unstructured":"Burns SM, Hulgaard H, Amon T, Borriello G (1995) An algorithm for exact bounds on the time separation of events in concurrent systems. IEEE Trans Comput 44(11):1306\u20131317. doi: 10.1109\/12.475126","journal-title":"IEEE Trans Comput"},{"key":"137_CR14","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/S0024-3795(02)00655-9","volume":"367","author":"P Butkovi\u010d","year":"2003","unstructured":"Butkovi\u010d P (2003) Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl 367:313\u2013335","journal-title":"Linear Algebra Appl"},{"key":"137_CR15","doi-asserted-by":"crossref","unstructured":"Butkovi\u010d P (2010) Max-linear systems: theory and algorithms. Springer","DOI":"10.1007\/978-1-84996-299-5"},{"issue":"4","key":"137_CR16","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1080\/026811199281967","volume":"14","author":"J Cochet-Terrasson","year":"1999","unstructured":"Cochet-Terrasson J, Gaubert S, Gunawardena J (1999) A constructive fixed-point theorem for min\u2013max functions. Dyn Stab Syst 14(4):407\u2013433","journal-title":"Dyn Stab Syst"},{"key":"137_CR17","doi-asserted-by":"crossref","unstructured":"Cohen G, Gaubert S, Nikoukhah R, Quadrat J (1991) Second order theory of min-linear systems and its application to discrete event systems. In: Proceedings of the 30th CDC, Brighton. doi: 10.1109\/CDC.1991.261654","DOI":"10.1109\/CDC.1991.261654"},{"key":"137_CR18","first-page":"105","volume-title":"Contemporary mathematics, vol 377","author":"G Cohen","year":"2005","unstructured":"Cohen G, Gaubert S, Quadrat JP, Singer I (2005) Max-plus convex sets and functions. In: Litvinov G, Maslov V (eds) Idempotent mathematics and mathematical physics. Contemporary mathematics, vol 377. AMS, Providence, pp 105\u2013129. arXiv:math\/0308166"},{"key":"137_CR19","volume-title":"Lecture notes in economics and mathematical systems, vol 166","author":"RA Cuninghame-Green","year":"1979","unstructured":"Cuninghame-Green RA (1979) Minimax algebra. In: Lecture notes in economics and mathematical systems, vol 166. Springer, Berlin"},{"key":"137_CR20","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0304-3975(02)00228-1","volume":"293","author":"RA Cuninghame-Green","year":"2003","unstructured":"Cuninghame-Green RA, Butkovi\u010d P (2003) The equation A\u2009\u2297\u2009x\u2009=\u2009B\u2009\u2297\u2009y over (max,+). Theor Comp Sci 293:3\u201312","journal-title":"Theor Comp Sci"},{"key":"137_CR21","doi-asserted-by":"crossref","unstructured":"Cuninghame-Green RA, Butkovi\u010d P (2008) Generalised eigenproblem in max algebra. In: Proceedings of the 9th international workshop WODES 2008, pp 236\u2013241. Available online from http:\/\/web.mat.bham.ac.uk\/P.Butkovic\/Grant.html (preprint)","DOI":"10.1109\/WODES.2008.4605951"},{"key":"137_CR22","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF01797235","volume":"6","author":"B De Schutter","year":"1996","unstructured":"De Schutter B, De Moor B (1996) A method to find all solutions of a system of multivariate polynomial equalities and inequalities in the max algebra. Discret Event Dyn Syst 6:115\u2013138","journal-title":"Discret Event Dyn Syst"},{"key":"137_CR23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4171\/dm\/154","volume":"9","author":"M Develin","year":"2004","unstructured":"Develin M, Sturmfels B (2004) Tropical convexity. Doc Math 9:1\u201327 (electronic). arXiv:math.MG\/0308254","journal-title":"Doc Math"},{"key":"137_CR24","doi-asserted-by":"crossref","unstructured":"Dhingra V, Gaubert S (2006) How to solve large scale deterministic games with mean payoff by policy iteration. In: Proceedings of the 1st international conference on performance evaluation methodologies and tools (VALUETOOLS), vol 180. Pisa, Italy. Article No. 12","DOI":"10.1145\/1190095.1190110"},{"issue":"I","key":"137_CR25","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/S0764-4442(97)82710-3","volume":"326","author":"S Gaubert","year":"1998","unstructured":"Gaubert S, Gunawardena J (1998a) The duality theorem for min\u2013max functions. C\u00a0R Acad Sci Paris 326(I):43\u201348","journal-title":"C\u00a0R Acad Sci Paris"},{"key":"137_CR26","volume-title":"Proc. of the fourth workshop on discrete event systems (WODES98)","author":"S Gaubert","year":"1998","unstructured":"Gaubert S, Gunawardena J (1998b) A non-linear hierarchy for discrete event dynamical systems. In: Proc. of the fourth workshop on discrete event systems (WODES98). IEE, Cagliari, Italy"},{"issue":"5\u20137","key":"137_CR27","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1016\/j.laa.2009.03.012","volume":"431","author":"S Gaubert","year":"2009","unstructured":"Gaubert S, Katz RD (2009) The tropical analogue of polar cones. Linear Algebra Appl 431(5\u20137):608\u2013625. arXiv:0805.3688","journal-title":"Linear Algebra Appl"},{"issue":"6","key":"137_CR28","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1007\/s10958-008-9243-8","volume":"155","author":"S Gaubert","year":"2008","unstructured":"Gaubert S, Sergeev S (2008) Cyclic projectors and separation theorems in idempotent convex geometry. J Math Sci 155(6):815\u2013829. arXiv:math\/0706.3347","journal-title":"J Math Sci"},{"key":"137_CR29","doi-asserted-by":"crossref","unstructured":"Gaubert S, Katz RD, Sergeev S (2011) Tropical linear programming and parametric mean-payoff games. J Symb Comput. arXiv:1101.3431 doi: 10.1016\/j.jsc.2011.12.049 (to appear)","DOI":"10.1016\/j.jsc.2011.12.049"},{"key":"137_CR30","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/BF01440235","volume":"4","author":"J Gunawardena","year":"1994","unstructured":"Gunawardena J (1994) Min-max functions. Discret Event Dyn Syst 4:377\u2013406","journal-title":"Discret Event Dyn Syst"},{"key":"137_CR31","doi-asserted-by":"crossref","unstructured":"Heidergott B, Olsder GJ, van der Woude J (2005) Max-plus at work. Princeton Univ. Press","DOI":"10.1515\/9781400865239"},{"key":"137_CR32","doi-asserted-by":"crossref","unstructured":"Kolokoltsov VN, Maslov VP (1997) Idempotent analysis and its applications. Kluwer Academic Pub.","DOI":"10.1007\/978-94-015-8901-7"},{"key":"137_CR33","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1137\/1011093","volume":"11","author":"TM Liggett","year":"1969","unstructured":"Liggett TM, Lippman SA (1969) Stochastic games with perfect information and time average payoff. SIAM Rev 11:604\u2013607","journal-title":"SIAM Rev"},{"issue":"5","key":"137_CR34","first-page":"758","volume":"69","author":"GL Litvinov","year":"2001","unstructured":"Litvinov GL, Maslov VP, Shpiz GB (2001) Idempotent functional analysis: an algebraic approach. Math Notes (Moscow) 69(5):758\u2013797. arXiv:math.FA\/0009128","journal-title":"Math Notes (Moscow)"},{"key":"137_CR35","doi-asserted-by":"crossref","first-page":"32","DOI":"10.13001\/1081-3810.1023","volume":"4","author":"JJ McDonald","year":"1998","unstructured":"McDonald JJ, Olesky DD, Schneider H, Tsatsomeros MJ, van den Driessche P (1998) Z-pencils. Electron J Linear Algebra 4:32\u201338","journal-title":"Electron J Linear Algebra"},{"key":"137_CR36","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.laa.2007.07.004","volume":"428","author":"V Mehrmann","year":"2008","unstructured":"Mehrmann V, Nabben R, Virnik E (2008) Generalization of Perron\u2013Frobenius theory to matrix pencils. Linear Algebra Appl 428:20\u201338","journal-title":"Linear Algebra Appl"},{"key":"137_CR37","unstructured":"Minc H (1988) Nonnegative matrices. Wiley"},{"issue":"2","key":"137_CR38","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1137\/S009753970037727X","volume":"33","author":"RH M\u00f6hring","year":"2004","unstructured":"M\u00f6hring RH, Skutella M, Stork F (2004) Scheduling with AND\/OR precedence constraints. SIAM J Comput 33(2):393\u2013415 (electronic). doi: 10.1137\/S009753970037727X","journal-title":"SIAM J Comput"},{"key":"137_CR39","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/0024-3795(86)90233-8","volume":"73","author":"R Nussbaum","year":"1986","unstructured":"Nussbaum R (1986) Convexity and log convexity for the spectral radius. Linear Algebra Appl 73:59\u2013122","journal-title":"Linear Algebra Appl"},{"key":"137_CR40","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF01805562","volume":"1","author":"G Olsder","year":"1991","unstructured":"Olsder G (1991) Eigenvalues of dynamic min\u2013max functions. Discret Event Dyn Syst 1:177\u2013207","journal-title":"Discret Event Dyn Syst"},{"key":"137_CR41","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1090\/conm\/495\/09704","volume-title":"Tropical and idempotent mathematics. Cont Math, vol 495","author":"S Sergeev","year":"2009","unstructured":"Sergeev S (2009) Multiorder, Kleene stars and cyclic projectors in the geometry of max cones. In: Litvinov GL, Sergeev SN (eds) Tropical and idempotent mathematics. Cont Math, vol 495. AMS, Providence, pp 317\u2013342. arXiv:0807.0921"},{"key":"137_CR42","unstructured":"Sergeev S (2010) Mean-payoff games and parametric tropical two-sided systems. University of Birmingham, School of Mathematics, Preprint 2010\/15. Available online from http:\/\/web.mat.bham.ac.uk\/P.Butkovic\/Grant.html"},{"issue":"5","key":"137_CR43","first-page":"715","volume":"47","author":"S Sergeev","year":"2011","unstructured":"Sergeev S (2011) On the problem Ax\u2009=\u2009\u03bbB x in max algebra: every system of intervals is a spectrum. Kybernetika 47(5):715\u2013721. arXiv:1001.4051","journal-title":"Kybernetika"},{"issue":"1\u20132","key":"137_CR44","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0304-3975(95)00188-3","volume":"158","author":"U Zwick","year":"1996","unstructured":"Zwick U, Paterson M (1996) The complexity of mean payoff games on graphs. Theor Comp Sci 158(1\u20132):343\u2013359","journal-title":"Theor Comp Sci"}],"container-title":["Discrete Event Dynamic Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10626-012-0137-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10626-012-0137-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10626-012-0137-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,25]],"date-time":"2023-06-25T09:44:11Z","timestamp":1687686251000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10626-012-0137-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,25]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,6]]}},"alternative-id":["137"],"URL":"https:\/\/doi.org\/10.1007\/s10626-012-0137-z","relation":{},"ISSN":["0924-6703","1573-7594"],"issn-type":[{"value":"0924-6703","type":"print"},{"value":"1573-7594","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,5,25]]}}}