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Comput."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>In the classic Minimum Bisection problem we are given as input an undirected graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \\leq 1$ and there are at most $k$ edges with one endpoint in $A$ and the other in $B$. In this paper we give an algorithm for Minimum Bisection with running time $2^{\\mathcal{O}(k^3)}n^3 \\log^3 n$. This is the first fixed parameter tractable algorithm for Minimum Bisection parameterized by $k$. At the core of our algorithm lies a new decomposition theorem that states that every graph $G$ can be decomposed by small separators into parts where each part is \u201chighly connected\u201d in the following sense: any separator of bounded size can separate only a limited number of vertices from each part of the decomposition. Our techniques generalize to the weighted setting, where we seek a bisection of minimum weight among solutions that contain at most $k$ edges.<\/jats:p>","DOI":"10.1137\/140988553","type":"journal-article","created":{"date-parts":[[2019,4,11]],"date-time":"2019-04-11T11:45:37Z","timestamp":1554983137000},"page":"417-450","source":"Crossref","is-referenced-by-count":18,"title":["Minimum Bisection Is Fixed-Parameter Tractable"],"prefix":"10.1137","volume":"48","author":[{"given":"Marek","family":"Cygan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Lokshtanov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marcin","family":"Pilipczuk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Micha\u0142","family":"Pilipczuk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saket","family":"Saurabh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,4,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1145\/210332.210337"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548302005369"},{"key":"atypb3","first-page":"76","author":"van Bevern R.","year":"2013","journal-title":"Computer Science"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00224-014-9557-5"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579448"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"T. N. Bui, C. Heigham, C. Jones, and F. T. Leighton,\n                      Improving the performance of the Kernighan-Lin and simulated annealing graph bisection algorithms\n                      , in Proceedings of the 26th ACM\/IEEE Design Automation Conference, 1989, pp. 775-778.","DOI":"10.1145\/74382.74527"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/0221016"},{"key":"atypb8","first-page":"43","author":"Bui T. N.","year":"2002","journal-title":"Proceedings of the Genetic and Evolutionary Computation Conference"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-014-2898-5"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-007-9130-6"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/15M1032077"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"M. Cygan, D. Lokshtanov, M. Pilipczuk, M. Pilipczuk, and S. Saurabh,\n                      Minimum Bisection is fixed parameter tractable\n                      , in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, 2014, pp. 323-332.","DOI":"10.1145\/2591796.2591852"},{"key":"atypb13","unstructured":"R. Diestel,\n                      Graph Theory\n                      , Springer, Berlin, 2012."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539701387660"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"U. Feige, R. Krauthgamer, and K. Nissim,\n                      Approximating the minimum bisection size (extended abstract)\n                      , in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 530-536.","DOI":"10.1145\/335305.335370"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"U. Feige and M. 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Thorup,\n                      The minimum $k$-way cut of bounded size is fixed-parameter tractable\n                      , in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, 2011, pp. 160-169.","DOI":"10.1109\/FOCS.2011.53"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1145\/2629614"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.10.007"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1145\/2500119"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/110855247"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/BF01758778"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"M. Naor, L. J. Schulman, and A. 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