{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:43:39Z","timestamp":1776836619285,"version":"3.51.2"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1-3","license":[{"start":{"date-parts":[[2008,3,1]],"date-time":"2008-03-01T00:00:00Z","timestamp":1204329600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/2.0"},{"start":{"date-parts":[[2008,3,5]],"date-time":"2008-03-05T00:00:00Z","timestamp":1204675200000},"content-version":"vor","delay-in-days":4,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/2.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2008,3]]},"DOI":"10.1007\/s00454-008-9050-5","type":"journal-article","created":{"date-parts":[[2008,3,5]],"date-time":"2008-03-05T00:41:56Z","timestamp":1204677716000},"page":"174-190","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":86,"title":["Generating All Vertices of a Polyhedron Is Hard"],"prefix":"10.1007","volume":"39","author":[{"given":"Leonid","family":"Khachiyan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Endre","family":"Boros","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Konrad","family":"Borys","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khaled","family":"Elbassioni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladimir","family":"Gurvich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2008,3,5]]},"reference":[{"key":"9050_CR1","unstructured":"Abdullahi, S.D.: Vertex enumeration and counting for certain classes of polyhedra. Ph.D. thesis, Computing (Computer Algorithms), Leeds University (2003)"},{"key":"9050_CR2","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, B., Seidel, R.: How good are convex hull algorithms. Comput. Geom. Theory Appl. 7, 265\u2013302 (1997)","journal-title":"Comput. Geom. Theory Appl."},{"issue":"3","key":"9050_CR3","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/BF02293050","volume":"8","author":"D. Avis","year":"1992","unstructured":"Avis, D., Fukuda, K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8(3), 295\u2013313 (1992)","journal-title":"Discrete Comput. Geom."},{"issue":"1\u20133","key":"9050_CR4","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/0166-218X(95)00026-N","volume":"65","author":"D. Avis","year":"1996","unstructured":"Avis, D., Fukudam, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1\u20133), 21\u201346 (1996)","journal-title":"Discrete Appl. Math."},{"key":"9050_CR5","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s10107-002-0363-5","volume":"95","author":"E. Amaldi","year":"2003","unstructured":"Amaldi, E., Pfetsch, M.E., Trotter, L.E.: On the maximum feasible subsystem problem, IISs and IIS-hypergraphs. Math. Program. 95, 533\u2013554 (2003)","journal-title":"Math. Program."},{"key":"9050_CR6","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1137\/0109008","volume":"9","author":"M.L. Balinski","year":"1961","unstructured":"Balinski, M.L.: An algorithm for finding all vertices of convex polyhedral sets. SIAM J. Appl. Math. 9, 72\u201381 (1961)","journal-title":"SIAM J. Appl. Math."},{"key":"9050_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1007\/978-3-540-25960-2_12","volume-title":"Integer Programming and Combinatorial Optimization, 10th International IPCO Conference","author":"E. Boros","year":"2004","unstructured":"Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Enumerating minimal dicuts and strongly connected subgraphs and related geometric problems. In: Bienstock, D., Nemhauser, G. (eds.) Integer Programming and Combinatorial Optimization, 10th International IPCO Conference. Lecture Notes in Computer Science, vol.\u00a03064, pp.\u00a0152\u2013162. Springer, Berlin (2004). (An extended version is to appear in Algorithmica)"},{"key":"9050_CR8","volume-title":"Hypergraphs","author":"C. Berge","year":"1989","unstructured":"Berge, C.: Hypergraphs. Elsevier-North Holland, Amsterdam (1989)"},{"key":"9050_CR9","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/PL00009389","volume":"20","author":"D. Bremner","year":"1998","unstructured":"Bremner, D., Fukuda, K., Marzetta, A.: Primal\u2013dual methods for vertex and facet enumeration. Discrete Comput. Geom. 20, 333\u2013357 (1998)","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"9050_CR10","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/S0925-7721(98)00021-2","volume":"11","author":"M.R. Bussieck","year":"1998","unstructured":"Bussieck, M.R., L\u00fcbbecke, M.E.: The vertex set of a 0\/1 polytope is strongly \u2118-enumerable. Comput. Geom. Theory Appl. 11(2), 103\u2013109 (1998)","journal-title":"Comput. Geom. Theory Appl."},{"key":"9050_CR11","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/PL00009410","volume":"21","author":"D. Bremner","year":"1999","unstructured":"Bremner, D.: Incremental convex hull algorothms are not output sensitive. Discrete Comput. Geom. 21, 57\u201368 (1999)","journal-title":"Discrete Comput. Geom."},{"key":"9050_CR12","volume-title":"An Introduction to Linear Programming","author":"A. Charnes","year":"1953","unstructured":"Charnes, A., Cooper, W.W., Henderson, A.: An Introduction to Linear Programming. Wiley, New\u00a0York (1953)"},{"key":"9050_CR13","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/BF02573985","volume":"10","author":"B. Chazelle","year":"1993","unstructured":"Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10, 377\u2013409 (1993)","journal-title":"Discrete Comput. Geom."},{"key":"9050_CR14","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/0377-2217(94)90152-X","volume":"73","author":"N. Chakravarti","year":"1994","unstructured":"Chakravarti, N.: Some results concerning post-infeasibility analysis. Eur. J. Oper. Res. 73, 139\u2013143 (1994)","journal-title":"Eur. J. Oper. Res."},{"key":"9050_CR15","volume-title":"Linear Programming","author":"V. Chv\u00e1tal","year":"1983","unstructured":"Chv\u00e1tal, V.: Linear Programming. Freeman, San Francisco (1983)"},{"issue":"1","key":"9050_CR16","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1145\/321556.321564","volume":"17","author":"D.R. Chand","year":"1970","unstructured":"Chand, D.R., Kapur, S.S.: An algorithm for convex polytopes. J. Assoc. Comput. Mach. 17(1), 78\u201386 (1970)","journal-title":"J. Assoc. Comput. Mach."},{"key":"9050_CR17","doi-asserted-by":"crossref","unstructured":"Cook, S.A.: The complexity of theorem proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151\u2013158 (1971)","DOI":"10.1145\/800157.805047"},{"key":"9050_CR18","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1287\/moor.8.3.381","volume":"8","author":"M.E. Dyer","year":"1983","unstructured":"Dyer, M.E.: The complexity of vertex enumeration methods. Math. Oper. Res. 8, 381\u2013402 (1983)","journal-title":"Math. Oper. Res."},{"key":"9050_CR19","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/BF01593771","volume":"12","author":"M.E. Dyer","year":"1977","unstructured":"Dyer, M.E., Proll, L.G.: An algorithm for determining all extreme points of a convex polytope. Math. Program. 12, 81\u201396 (1977)","journal-title":"Math. Program."},{"key":"9050_CR20","doi-asserted-by":"publisher","first-page":"1278","DOI":"10.1137\/S0097539793250299","volume":"24","author":"T. Eiter","year":"1995","unstructured":"Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24, 1278\u20131304 (1995)","journal-title":"SIAM J. Comput."},{"key":"9050_CR21","first-page":"1","volume":"124","author":"J. Farkas","year":"1901","unstructured":"Farkas, J.: Theorie der einfachen ungleichungen. J. Rein. Angew. Math. 124, 1\u201327 (1901)","journal-title":"J. Rein. Angew. Math."},{"key":"9050_CR22","doi-asserted-by":"publisher","first-page":"618","DOI":"10.1006\/jagm.1996.0062","volume":"21","author":"M. Fredman","year":"1996","unstructured":"Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21, 618\u2013628 (1996)","journal-title":"J. Algorithms"},{"key":"9050_CR23","first-page":"345","volume":"5","author":"R.W. Floyd","year":"1962","unstructured":"Floyd, R.W.: Algorithm 97: Shortest path. Commun. Assoc. Comput. Mach. 5, 345 (1962)","journal-title":"Commun. Assoc. Comput. Mach."},{"key":"9050_CR24","first-page":"1","volume":"8","author":"K. Fukuda","year":"1997","unstructured":"Fukuda, K., Liebling, Th.M., Margot, F.: Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron. CGTA 8, 1\u201312 (1997)","journal-title":"CGTA"},{"key":"9050_CR25","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/BF02020271","volume":"9","author":"T. Gallai","year":"1958","unstructured":"Gallai, T.: Maximum-minimum S\u00e4tze \u00fcber Graphen. Acta Math. Acad. Sci. Hung. 9, 395\u2013434 (1958)","journal-title":"Acta Math. Acad. Sci. Hung."},{"issue":"1","key":"9050_CR26","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1287\/ijoc.2.1.61","volume":"2","author":"J. Gleeson","year":"1990","unstructured":"Gleeson, J., Ryan, J.: Identifying minimally infeasible subsystems of inequalities. ORSA J. Comput. 2(1), 61\u201363 (1990)","journal-title":"ORSA J. Comput."},{"key":"9050_CR27","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/0304-3975(78)90006-3","volume":"6","author":"D.S. Johnson","year":"1978","unstructured":"Johnson, D.S., Preparata, F.P.: The densest hemisphere problem. Theor. Comput. Sci. 6, 93\u2013107 (1978)","journal-title":"Theor. Comput. Sci."},{"key":"9050_CR28","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/0020-0190(88)90065-8","volume":"27","author":"D.S. Johnson","year":"1988","unstructured":"Johnson, D.S., Papadimitriou, Ch.H.: On generating all maximal independent sets. Inf. Process. Lett. 27, 119\u2013123 (1988)","journal-title":"Inf. Process. Lett."},{"key":"9050_CR29","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/0304-3975(86)90174-X","volume":"44","author":"M.R. Jerrum","year":"1986","unstructured":"Jerrum, M.R., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 44, 169\u2013188 (1986)","journal-title":"Theor. Comput. Sci."},{"key":"9050_CR30","first-page":"191","volume":"20","author":"L. Khachiyan","year":"1979","unstructured":"Khachiyan, L.: A polynomial algorithm in linear programming. Sov. Math. Dokl. 20, 191\u2013194 (1979)","journal-title":"Sov. Math. Dokl."},{"key":"9050_CR31","unstructured":"Lov\u00e1sz, L.: Combinatorial optimization: some problems and trends. DIMACS Technical Report 92-53, Rutgers University (1992)"},{"key":"9050_CR32","doi-asserted-by":"publisher","first-page":"558","DOI":"10.1137\/0209042","volume":"9","author":"E. Lawler","year":"1980","unstructured":"Lawler, E., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9, 558\u2013565 (1980)","journal-title":"SIAM J. Comput."},{"key":"9050_CR33","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1287\/opre.21.1.247","volume":"21","author":"T.H. Mattheiss","year":"1973","unstructured":"Mattheiss, T.H.: An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities. Oper. Res. 21, 247\u2013260 (1973)","journal-title":"Oper. Res."},{"key":"9050_CR34","doi-asserted-by":"crossref","unstructured":"Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The double description method. In: H.W.\u00a0Kuhn and A.W.\u00a0Tucker (eds.) Contributions to the Theory of Games, vol.\u00a0II, pp.\u00a051\u201373 (1953)","DOI":"10.1515\/9781400881970-004"},{"key":"9050_CR35","unstructured":"Pfetsch, M.E.: The maximum feasible subsystem problem and vertex-facet incidences of polyhedra. Dissertation, TU Berlin (2002)"},{"issue":"1","key":"9050_CR36","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/BF01582058","volume":"63","author":"J.S. Provan","year":"1994","unstructured":"Provan, J.S.: Efficient enumeration of the vertices of polyhedra associated with network lp\u2019s. Math. Program. 63(1), 47\u201364 (1994)","journal-title":"Math. Program."},{"key":"9050_CR37","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1002\/net.1975.5.3.237","volume":"5","author":"R.C. Read","year":"1975","unstructured":"Read, R.C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5, 237\u2013252 (1975)","journal-title":"Networks"},{"issue":"4","key":"9050_CR38","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1137\/S0895480194270469","volume":"9","author":"J. Ryan","year":"1996","unstructured":"Ryan, J.: IIS-hypergraphs. SIAM J. Discrete Math. 9(4), 643\u2013653 (1996)","journal-title":"SIAM J. Discrete Math."},{"key":"9050_CR39","volume-title":"Theory of Linear and Integer Programming","author":"A. Schrijver","year":"1986","unstructured":"Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)"},{"key":"9050_CR40","volume-title":"Combinatorial Optimization: Polyhedra and Efficiency, vol.\u00a0A","author":"A. Schrijver","year":"2003","unstructured":"Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol.\u00a0A. Springer, Berlin (2003)"},{"key":"9050_CR41","volume-title":"Output-size sensitive algorithms for constructive problems in computational geometry. Computer Science","author":"R. Seidel","year":"1986","unstructured":"Seidel, R.: Output-size sensitive algorithms for constructive problems in computational geometry. Computer Science. Cornell University, Ithaka (1986)"},{"key":"9050_CR42","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0196-6774(85)90017-3","volume":"6","author":"G. Swart","year":"1985","unstructured":"Swart, G.: Finding the convex hull facet by facet. J. Algorithms 6, 17\u201348 (1985)","journal-title":"J. Algorithms"},{"key":"9050_CR43","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1137\/0208032","volume":"8","author":"L.G. Valiant","year":"1979","unstructured":"Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410\u2013421 (1979)","journal-title":"SIAM J. Comput."},{"key":"9050_CR44","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1145\/321105.321107","volume":"9","author":"S. Warshall","year":"1962","unstructured":"Warshall, S.: A theorem on boolean matrices. J. Assoc. Comput. Mach. 9, 11\u201312 (1962)","journal-title":"J. Assoc. Comput. Mach."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-008-9050-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-008-9050-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-008-9050-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-008-9050-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T19:24:47Z","timestamp":1630437887000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-008-9050-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":44,"journal-issue":{"issue":"1-3","published-print":{"date-parts":[[2008,3]]}},"alternative-id":["9050"],"URL":"https:\/\/doi.org\/10.1007\/s00454-008-9050-5","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3]]},"assertion":[{"value":"21 July 2005","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 June 2006","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2008","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}