{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:26:32Z","timestamp":1787232392494,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["05M14ZAM"],"award-info":[{"award-number":["05M14ZAM"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper, we present a new, optimization-based method to exhibit cyclic behavior in nonreversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing nonreversible biological processes, in particular, molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring oscillator with three genes, we show that this approach can detect cycles that have a productivity one magnitude larger than classical spectral analysis methods. Our method applies to general nonequilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.<\/jats:p>","DOI":"10.1137\/16m1091162","type":"journal-article","created":{"date-parts":[[2018,2,15]],"date-time":"2018-02-15T13:44:40Z","timestamp":1518702280000},"page":"248-265","source":"Crossref","is-referenced-by-count":3,"title":["Mixed-Integer Programming for Cycle Detection in Nonreversible Markov Processes"],"prefix":"10.1137","volume":"16","author":[{"given":"Jakob","family":"Witzig","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Isabel","family":"Beckenbach","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leon","family":"Eifler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Konstantin","family":"Fackeldey","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ambros","family":"Gleixner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas","family":"Grever","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marcus","family":"Weber","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,15]]},"reference":[{"key":"atypb1","first-page":"6","author":"Achterberg T.","year":"2008","journal-title":"Berlin"},{"key":"atypb2","unstructured":"D. Applegate, R. Bixby, W. Cook, and V. Chva\u0301tal,\n                      On the Solution of Travelling Salesman Problems\n                      , Technical report CRPC-TR98744, Rice University, Houston, TX, 1998."},{"key":"atypb3","unstructured":"T. Berthold,\n                      Heuristic Algorithms in Global MINLP Solvers\n                      , Ph.D. thesis, Technical University of Berlin, Berlin, 2014,http:\/\/www.zib.de\/berthold\/Berthold2014.pdf."},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"G. Bowman, V. Pande, and F. Noe\u0301, eds.\n                      An Introduction to Markov State Models and their Application to Long Timescale Molecular Simulation\n                      , Adv. Exp. Med. Biol. 797, Springer, Berlin, 2014.","DOI":"10.1007\/978-94-007-7606-7"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"S. Brooks, A. Gelman, G. Jones, and X.L. Meng,\n                      Handbook of Markov Chain Monte Carlo\n                      , CRC Press, 2011.","DOI":"10.1201\/b10905"},{"key":"atypb6","first-page":"241","author":"Dahlhaus E.","year":"1992","journal-title":"New York"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/8.3.250"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996313002"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00095-1"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.10.026"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.3934\/jcd.2015.2.1"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/15M1032272"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1038\/35002125"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/490\/1\/012016"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"M. Fischetti and A. Lodi,\n                      Heuristics in mixed integer programming\n                      , in Wiley Encyclopedia of Operations Research and Management Science, J. J. Cochran, L. A. Cox, P. Keskinocak, J. P. Kharoufeh, and J. C. Smith, eds., Wiley, Hoboken, NJ, 2010, eorms0376.","DOI":"10.1002\/9780470400531.eorms0376"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/BF03006558"},{"key":"atypb17","first-page":"15","volume":"2","author":"Gamrath G.","journal-title":"The SCIP Optimization Suite 3"},{"key":"atypb18","unstructured":"M. R. Garey and D. S. Johnson,\n                      Computers and Intractability: A Guide to the Theory of NP-Completeness\n                      , Freeman, New York, 1979."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1109\/43.159993"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1970.tb01770.x"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"A. H. Land and A. G. Doig,\n                      An automatic method of solving discrete programming problems\n                      , Econometrica, (1960), pp. 497-520.","DOI":"10.2307\/1910129"},{"key":"atypb22","unstructured":"R. Lidl and H. Niederreiter,\n                      Finite Fields\n                      , Cambridge University Press, Cambridge, 1984."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/0022-314X(88)90025-X"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1967"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1063\/1.3565032"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1287\/ijoc.6.4.445"},{"key":"atypb27","unstructured":"A. Schrijver,\n                      Combinatorial Optimization: Polyhedra and Efficiency\n                      , Algorithms Combin. 24, Springer, Berlin, 2003."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1109\/34.868688"},{"key":"atypb29","unstructured":"U. von Luxburg,\n                      Tutorial on Spectral Clustering\n                      , Max Planck Institute for Biological Cybernetics, Technical report 149, Tu\u0308ubingen, Germany, 2006."},{"key":"atypb30","unstructured":"M. Weber,\n                      A Subspace Approach to Molecular Markov State Models via a New Infinitesimal Generator\n                      , habilitation thesis, Freie Universita\u0308t Berlin, Berlin, 2011."},{"key":"atypb31","unstructured":"M. Weber,\n                      Eigenvalues of Non-Reversible Markov Chains - A Case Study\n                      , Technical report 17-13, Zuse Institute Berlin, Berlin, 2017."},{"key":"atypb32","unstructured":"M. Weber and K. Fackeldey,\n                      G-PCCA: Spectral Clustering for Non-Reversible Markov Chains\n                      , ZIB-Report 15-35, Zuse Institute Berlin, Berlin, 2015."}],"container-title":["Multiscale Modeling &amp; Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1091162","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:45:58Z","timestamp":1787229958000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1091162"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M1091162"],"URL":"https:\/\/doi.org\/10.1137\/16m1091162","relation":{},"ISSN":["1540-3459","1540-3467"],"issn-type":[{"value":"1540-3459","type":"print"},{"value":"1540-3467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}