{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:24:53Z","timestamp":1787340293122,"version":"build-2736575974"},"reference-count":40,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>Let $G=(V,E)$ be an edge\u2010weighted undirected graph with n vertices and m edges. We present a deterministic algorithm to compute a minimum k\u2010way cut of G for a given k. Our algorithm is a divide\u2010and\u2010conquer method based on a procedure that reduces an instance of the minimum k\u2010way cut problem to $O(n^{2k-5})$ instances of the minimum $(\\lfloor (k+\\sqrt{k})\/2\\rfloor+1)$\u2010way cut problem, and can be implemented to run in $O(n^{4k\/(1-1.71\/\\sqrt{k}) -31} )$ time. With a slight modification, the algorithm can find all minimum k\u2010way cuts in $O(n^{4k\/(1-1.71\/\\sqrt{k}) -16} )$ time.<\/jats:p>","DOI":"10.1137\/050631616","type":"journal-article","created":{"date-parts":[[2006,12,26]],"date-time":"2006-12-26T11:06:15Z","timestamp":1167131175000},"page":"1329-1341","source":"Crossref","is-referenced-by-count":36,"title":["A Deterministic Algorithm for Finding All Minimum\n                    <i>k<\/i>\n                    \u2010Way Cuts"],"prefix":"10.1137","volume":"36","author":[{"given":"Yoko","family":"Kamidoi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Noriyoshi","family":"Yoshida","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hiroshi","family":"Nagamochi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,12,21]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-9260(95)00008-4"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0037(199909)34:2<102::AID-NET3>3.0.CO;2-X"},{"key":"R3","unstructured":"C. J. Colbourn,\n                      The Combinatorics of Network Reliability\n                      , Oxford University Press, Oxford, UK, 1987."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1145\/3828.3829"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792225297"},{"key":"R6","unstructured":"W. E. Donath,\n                      Logic partitioning\n                      , in Physical Design Automation of VLSI Systems, B. T. Preas and M. J. Lorenzetti, eds., Benjamin Cummings, Menlo Park, CA, 1988, pp. 65\u201386."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1145\/48014.61051"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1287\/moor.19.1.24"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(83)90031-5"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"H. W. Hamacher, J.\u2010C. Picard, and M. Queyranne,\n                      Ranking the cuts and cut\u2010sets of a network\n                      , in Algebraic and Combinatorial Methods in Operations Research, North\u2013Holland Math. Stud. 95, North\u2013Holland, Amsterdam, 1984, pp. 183\u2013200.","DOI":"10.1016\/S0304-0208(08)72962-1"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(84)90083-X"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1994.1043"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1993.1033"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(98)00036-5"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(91)90021-P"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"S. Kapoor,\n                      \n                        On minimum 3\u2010cuts and approximating\n                        k\n                        \u2010cuts using cut trees\n                      \n                      , in Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 1084, Springer\u2010Verlag, Berlin, 1996, pp. 132\u2013146.","DOI":"10.1007\/3-540-61310-2_11"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"Y. Kamidoi, S. Wakabayashi, and N. Yoshida,\n                      \n                        Faster algorithms for finding a minimum\n                        k\n                        \u2010way cut in a weighted graph\n                      \n                      , in Proceedings of the IEEE International Symposium on Circuits and Systems, IEEE Circuits and Systems Society, Piscataway, NJ, 1997, pp. 1009\u20131012.","DOI":"10.1109\/ISCAS.1997.621902"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-001-0070-2"},{"key":"R19","unstructured":"D. R. Karger,\n                      Global min\u2010cuts in RNC, and other ramifications of a simple min\u2010cut algorithm\n                      , in Proceedings of the ACM\u2013SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 1993, pp. 21\u201330."},{"key":"R20","doi-asserted-by":"crossref","unstructured":"D. R. Karger,\n                      Minimum cuts in near\u2010linear time\n                      , in Proceedings of the 28th ACM Symposium on Theory of Computing, ACM, New York, 1996, pp. 56\u201363.","DOI":"10.1145\/237814.237829"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"D. R. Karger, P. Klein, C. Stein, M. Thorup, and N. Young,\n                      Rounding algorithms for a geometric embedding of minimum multiway cut\n                      , in Proceedings of the 31st ACM Symposium on Theory of Computing, ACM, New York, 1999, pp. 668\u2013678.","DOI":"10.1145\/301250.301430"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"D. R. Karger and C. Stein,\n                      An $\\tilde{O}(n^2)$ algorithm for minimum cuts\n                      , in Proceedings of the 25th ACM Symposium on Theory of Computing, ACM, New York, 1993, pp. 757\u2013765.","DOI":"10.1145\/167088.167281"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1145\/234533.234534"},{"key":"R24","first-page":"434","volume":"15","author":"Karzanov A. V.","year":"1974","journal-title":"Soviet Math. Dokl.","ISSN":"https:\/\/id.crossref.org\/issn\/0197-6788","issn-type":"print"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"C. H. Lee, M. Kim, and C. I. Park,\n                      \n                        An efficient\n                        k\n                        \u2010way graph partitioning algorithm for task allocation in parallel computing systems\n                      \n                      , in Proceedings of the IEEE International Conference on Computer\u2010Aided Design, IEEE Computer Society, Los Alamitos, CA, 1990, pp. 748\u2013751.","DOI":"10.1109\/ICSI.1990.138741"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"T. Lengaur,\n                      Combinatorial Algorithms for Integrated Circuit Layout\n                      , Wiley, New York, 1990.","DOI":"10.1007\/978-3-322-92106-2"},{"key":"R27","unstructured":"M. S. Levine,\n                      Faster randomized algorithms for computing minimum $\\{3,4,5,6\\}$\u2010way cuts\n                      , in Proceedings of the ACM\u2013SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2000, pp. 735\u2013742."},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/0405004"},{"key":"R29","doi-asserted-by":"crossref","unstructured":"H. Nagamochi and T. Ibaraki,\n                      A fast algorithm for computing minimum 3\u2010way and 4\u2010way cuts\n                      , in Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 1610, Springer\u2010Verlag, 1999, pp. 377\u2013390.","DOI":"10.1007\/3-540-48777-8_28"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/PL00011383"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009804919645"},{"key":"R32","unstructured":"J. Naor and Y. Rabani,\n                      \n                        Tree packing and approximating\n                        k\n                        \u2010cuts\n                      \n                      , in Proceedings of the ACM\u2013SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2001, pp. 26\u201327."},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792251730"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1109\/TSE.1977.233840"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(99)00092-X"},{"key":"R36","doi-asserted-by":"crossref","unstructured":"V. Vazirani and M. Yannakakis,\n                      Suboptimal cuts: Their enumeration, weight, and number\n                      , in Algebraic and Logic Programming, Lecture Notes in Comput. Sci. 632, Springer\u2010Verlag, Berlin, 1992, pp. 366\u2013377.","DOI":"10.1007\/3-540-55719-9_88"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.2000.1148"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011620607786"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2003.10.007"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-004-0510-2"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050631616","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:42Z","timestamp":1787336922000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050631616"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,12,21]]},"references-count":40,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/050631616"],"URL":"https:\/\/doi.org\/10.1137\/050631616","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,12,21]]}}}