{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,20]],"date-time":"2026-09-20T12:20:39Z","timestamp":1789906839027,"version":"4.0.1"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. Cryan and Dyer [J. Comput. System Sci., 67 (2003), pp. 291-310] have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which employs Markov chain methods indirectly. They leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we show that the \"2 x 2 heat-bath\" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris [Random Walks in Convex Sets, Ph.D. thesis, Department of Statistics, University of California, Berkeley, CA, 2000] and Morris and Sinclair [SIAM J. Comput., 34 (2004), pp. 195-226] for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste [Ann. Appl. Probab., 3 (1993), pp. 696-730] to show that the 2 x 2 chain is also rapidly mixing.<\/jats:p>","DOI":"10.1137\/s0097539703434243","type":"journal-article","created":{"date-parts":[[2006,8,24]],"date-time":"2006-08-24T15:29:50Z","timestamp":1156433390000},"page":"247-278","source":"Crossref","is-referenced-by-count":19,"title":["Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows"],"prefix":"10.1137","volume":"36","author":[{"given":"Mary","family":"Cryan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Dyer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leslie Ann","family":"Goldberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Jerrum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Russell","family":"Martin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,27]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1287\/moor.19.4.769"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199608\/09)9:1\/2<55::AID-RSA4>3.0.CO;2-A"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(03)00014-X"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-003-0383-9"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176349634"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"P. Diaconis and A. Gangolli,\n                      Rectangular arrays with fixed margins\n                      , in Discrete Probability and Algorithms, IMA Vol. Math. Appl. 72, D. Aldous, P. P. Varaiya, J. Spencer, and J. M. Steele, eds., Springer\u2010Verlag, New York, 1995, pp. 15\u201341.","DOI":"10.1007\/978-1-4612-0801-3_3"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177005359"},{"key":"R8","unstructured":"P. Diaconis and L. Saloff\u2010Coste,\n                      Random Walk on Contingency Tables with Fixed Row and Column Sums\n                      , Technical report, Department of Mathematics, Harvard University, Cambridge, MA, 1995."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177005980"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"M. Dyer,\n                      Approximate counting by dynamic programming\n                      , in Proceedings of the 35th ACM Symposium on Theory of Computing, San Diego, 2003, pp. 693\u2013699.","DOI":"10.1145\/780542.780643"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(99)00136-X"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199707)10:4<487::AID-RSA4>3.0.CO;2-Q"},{"key":"R13","volume-title":"Probability and random processes","author":"Grimmett G.","year":"1992"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01086559"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199808)13:1<71::AID-RSA4>3.0.CO;2-P"},{"key":"R16","unstructured":"M. Jerrum and A. Sinclair,\n                      The Markov chain Monte Carlo method: An approach to approximate counting and integration\n                      , in Approximation Algorithms for NP\u2010Hard Problems, D. S. Hochbaum, ed., PWS, Boston, 1996, pp. 482\u2013520."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(86)90174-X"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"B. Morris,\n                      Improved bounds for sampling contingency tables\n                      , in 3rd International Workshop on Randomization and Approximation Techniques in Computer Science, Lecture Notes in Comput. Sci. 1671, Springer\u2010Verlag, New York, 1999, pp. 121\u2013129.","DOI":"10.1007\/978-3-540-48413-4_12"},{"key":"R19","unstructured":"B. Morris,\n                      Random Walks in Convex Sets\n                      , Ph.D. thesis, Department of Statistics, University of California, Berkeley, CA, 2000."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702411915"},{"key":"R21","unstructured":"J. Mount,\n                      Application of Convex Sampling to Optimization and Contingency Table Generation\n                      , Ph.D. thesis, Technical report CMU\u2010CS\u201095\u2010152, Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, 1995."},{"key":"R22","doi-asserted-by":"crossref","unstructured":"I. Pak,\n                      On sampling integer points in polyhedra\n                      , in Foundations of Computational Mathematics, World Scientific, River Edge, NJ, 2002, pp. 319\u2013324.","DOI":"10.1142\/9789812778031_0013"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1063\/1.533199"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300000390"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1913.143.128"},{"key":"R26","volume-title":"Approximation algorithms","author":"Vazirani Vijay","year":"2001"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1063\/1.533196"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539703434243","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:25:00Z","timestamp":1787336700000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539703434243"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/S0097539703434243"],"URL":"https:\/\/doi.org\/10.1137\/s0097539703434243","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}