{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T02:48:48Z","timestamp":1785293328803,"version":"3.55.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,6,16]],"date-time":"2015-06-16T00:00:00Z","timestamp":1434412800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Lendulet"},{"name":"TAMOP","award":["4.2.2.C-11\/1\/KONV-2012-0001"],"award-info":[{"award-number":["4.2.2.C-11\/1\/KONV-2012-0001"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2016,1]]},"DOI":"10.1007\/s10589-015-9760-6","type":"journal-article","created":{"date-parts":[[2015,6,15]],"date-time":"2015-06-15T03:29:54Z","timestamp":1434338994000},"page":"45-67","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Using multiobjective optimization to map the entropy region"],"prefix":"10.1007","volume":"63","author":[{"given":"L\u00e1szl\u00f3","family":"Csirmaz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2015,6,16]]},"reference":[{"issue":"5\u20136","key":"9760_CR1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/S0925-7721(96)00023-5","volume":"7","author":"D Avis","year":"1997","unstructured":"Avis, D., Bremner, D., Seidel, R.: How good are convex hull algorithms? Comput. Geom. 7(5\u20136), 265\u2013301 (1997)","journal-title":"Comput. Geom."},{"key":"9760_CR2","doi-asserted-by":"crossref","unstructured":"Baber, R., Christofides, D., Dang, A.N., Riis, S., Vaughan, E.R.: Multiple unicasts, graph guessing games, and non-Shannon inequalities. In: Proc. NetCod 2013, Calgary, pp. 1\u20136","DOI":"10.1109\/NetCod.2013.6570823"},{"issue":"4","key":"9760_CR3","doi-asserted-by":"crossref","first-page":"1950","DOI":"10.1109\/SURV.2013.013013.00104","volume":"15","author":"R Bassoli","year":"2013","unstructured":"Bassoli, R., Marques, H., Rodriguez, J., Shum, K.W., Tafazolli, R.: Network coding theory: a survey. IEEE Commun. Surveys Tutor. 15(4), 1950\u20131978 (2013)","journal-title":"IEEE Commun. Surveys Tutor."},{"key":"9760_CR4","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/978-3-642-20901-7_2","volume-title":"Coding and Cryptology. LNCS","author":"A Beimel","year":"2011","unstructured":"Beimel, A.: Secret-sharing schemes: a survey. Coding and Cryptology. LNCS, pp. 11\u201346. Springer, Heidelberg (2011)"},{"issue":"1","key":"9760_CR5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1008215702611","volume":"13","author":"P Benson","year":"1998","unstructured":"Benson, P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear program. J. Glob. Optim. 13(1), 1\u201324 (1998)","journal-title":"J. Glob. Optim."},{"key":"9760_CR6","unstructured":"Bremner, D.: On the complexity of vertex and facet enumeration for convex polytopes. PhD Thesis, School of Computer Science, McGill University (1997)"},{"key":"9760_CR7","unstructured":"Burton, B.A., Ozlen, M.: Projective geometry and the outer approximation algorithm for multiobjective linear programming, arXiv:1006.3085 (2010)"},{"key":"9760_CR8","doi-asserted-by":"crossref","first-page":"3261","DOI":"10.1109\/TIT.2003.820037","volume":"49","author":"TH Chan","year":"2003","unstructured":"Chan, T.H.: Balanced information inequalities. IEEE Trans. Inform. Theory 49, 3261\u20133267 (2003)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9760_CR9","doi-asserted-by":"crossref","first-page":"379","DOI":"10.3390\/e13020379","volume":"13","author":"TH Chan","year":"2011","unstructured":"Chan, T.H.: Recent progresses in characterising information inequalities. Entropy 13, 379\u2013401 (2011)","journal-title":"Entropy"},{"key":"9760_CR10","doi-asserted-by":"crossref","first-page":"6811","DOI":"10.1109\/TIT.2014.2352273","volume":"60","author":"L Csirmaz","year":"2014","unstructured":"Csirmaz, L.: Book inequalities. IEEE Trans. Inform. Theory 60, 6811\u20136818 (2014)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9760_CR11","unstructured":"Dougherty, R., Freiling, C., Zeger, K.: Non-Shannon information inequalities in four random variables, arXiv:1104.3602 (2011)"},{"key":"9760_CR12","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s10898-011-9709-y","volume":"52","author":"M Ehrgott","year":"2012","unstructured":"Ehrgott, M., L\u00f6hne, A., Shao, L.: A dual variant of Benson\u2019s outer approximation algorithm for multiple objective linear programming. J. Glob. Optim. 52, 757\u2013778 (2012)","journal-title":"J. Glob. Optim."},{"key":"9760_CR13","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/3-540-61576-8_77","volume-title":"Combinatorics and Computer Science (Brest, 1995). LNCS","author":"K Fukuda","year":"1996","unstructured":"Fukuda, K., Prodon, A.: Double description method revisited. Combinatorics and Computer Science (Brest, 1995). LNCS, vol. 1120, pp. 91\u2013111. Springer, Berlin (1996)"},{"key":"9760_CR14","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s10898-013-0098-2","volume":"59","author":"AH Hamel","year":"2013","unstructured":"Hamel, A.H., L\u00f6hne, A., Rudloff, B.: Benson type algorithms for linear vector optimization and applications. J. Glob. Optim. 59, 811\u2013836 (2013)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"9760_CR15","doi-asserted-by":"crossref","first-page":"836","DOI":"10.1137\/060674831","volume":"19","author":"F Heyde","year":"2008","unstructured":"Heyde, F., L\u00f6hne, A.: Geometric duality in multiple objective linear programming. SIAM J. Optim. 19(2), 836\u2013845 (2008)","journal-title":"SIAM J. Optim."},{"key":"9760_CR16","doi-asserted-by":"crossref","unstructured":"Kaced, T.: Equivalence of two proof techniques for non-Shannon-type inequalities, In: Proceedings of the 2013 IEEE International Symposium on Information Theory, Istambul, pp. 236\u2013240 (2013)","DOI":"10.1109\/ISIT.2013.6620223"},{"key":"9760_CR17","doi-asserted-by":"crossref","unstructured":"Madiman, M., Marcus, A.W., Tetali, P.: Information-theoretic inequalities in additive combinatorics. In: IEEE ITW, pp. 1\u20134 (2010)","DOI":"10.1109\/ITWKSPS.2010.5503129"},{"issue":"2","key":"9760_CR18","doi-asserted-by":"crossref","first-page":"147","DOI":"10.4310\/CIS.2002.v2.n2.a3","volume":"2","author":"K Makarychev","year":"2002","unstructured":"Makarychev, K., Makarychev, Yu., Romashchenko, A., Vereshchagin, N.: A new class of non-Shannon-type inequalities for entropies. Commun. Inf. Syst. 2(2), 147\u2013166 (2002)","journal-title":"Commun. Inf. Syst."},{"key":"9760_CR19","doi-asserted-by":"crossref","unstructured":"Matus, F.: Infinitely many information inequalities. In: Proceedings ISIT, pp. 41\u201347, 24\u201329 June 2007. Nice, France","DOI":"10.1109\/ISIT.2007.4557201"},{"issue":"1","key":"9760_CR20","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1109\/TIT.2006.887090","volume":"53","author":"F Matus","year":"2007","unstructured":"Matus, F.: Two constructions on limits of entropy functions. IEEE Trans. Inform. Theory 53(1), 320\u2013330 (2007)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9760_CR21","unstructured":"Matus, F.: Personal communication (2012)"},{"key":"9760_CR22","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1017\/S0963548300001644","volume":"4","author":"F Matus","year":"1995","unstructured":"Matus, F., Studeny, M.: Conditional independencies among four random variables I. Comb. Probab. Comput. 4, 269\u2013278 (1995)","journal-title":"Comb. Probab. Comput."},{"issue":"4","key":"9760_CR23","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1137\/0202023","volume":"2","author":"WB McRae","year":"1973","unstructured":"McRae, W.B., Davidson, E.R.: An algorithm for the extreme rays of a pointed convex polyhedral cone. SIAM J. Comput. 2(4), 281\u2013293 (1973)","journal-title":"SIAM J. Comput."},{"key":"9760_CR24","unstructured":"Pippenger, N.: What are the laws of information theory. In: Special Problems on Communication and Computation Conference. Palo Alt, California, 3\u20135 Sept 1986"},{"key":"9760_CR25","volume-title":"Probabilistic Conditional Independence Structures","author":"M Studen\u00fd","year":"2005","unstructured":"Studen\u00fd, M.: Probabilistic Conditional Independence Structures. Springer, New York (2005)"},{"key":"9760_CR26","doi-asserted-by":"crossref","unstructured":"MacLaren Walsh, J., Weber, S.: Relationships among bounds for the region of entropic vectors in four variables. In: Allerton Conference on Communication, Control, and Computing (2010)","DOI":"10.1109\/ALLERTON.2010.5707066"},{"key":"9760_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-8608-5","volume-title":"A First Course in Information Theory","author":"RW Yeung","year":"2002","unstructured":"Yeung, R.W.: A First Course in Information Theory. Kluwer Academic\/Plenum Publishers, New York (2002)"},{"issue":"4","key":"9760_CR28","doi-asserted-by":"crossref","first-page":"1440","DOI":"10.1109\/18.681320","volume":"44","author":"Z Zhang","year":"1998","unstructured":"Zhang, Z., Yeung, R.W.: On characterization of entropy function via information inequalities. Proc. IEEE Trans. Inf. Theory 44(4), 1440\u20131452 (1998)","journal-title":"Proc. IEEE Trans. Inf. Theory"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-015-9760-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-015-9760-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-015-9760-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,13]],"date-time":"2022-05-13T03:59:15Z","timestamp":1652414355000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-015-9760-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,16]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["9760"],"URL":"https:\/\/doi.org\/10.1007\/s10589-015-9760-6","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,16]]}}}