{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:44:47Z","timestamp":1787323487658,"version":"3.56.0"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Given a graph $G=(V,E)$ with node weights $\\varphi_v \\in \\mathbb{N}\\cup\\{0\\}$, $v\\in V$, and some number $F\\in \\mathbb{N}\\cup\\{0\\}$, the convex hull of the incidence vectors of all cuts $\\delta(S)$, $S\\subseteq V$, with $\\varphi(S)\\le F$ and $\\varphi(V\\setminus S)\\le F$ is called the bisection cut polytope. We study the facial structure of this polytope which shows up in many graph partitioning problems with applications in VLSI design or frequency assignment. We give necessary and in some cases sufficient conditions for the knapsack tree inequalities introduced in [C. E. Ferreira et al., Math. Programming, 74 (1996), pp. 247\u2013267] to be facet-defining. We extend these inequalities to a richer class by exploiting the fact that each cut intersects each cycle in an even number of edges. Finally, we present a new class of inequalities that are based on nonconnected substructures yielding nonlinear right-hand sides. We show that the supporting hyperplanes of the convex envelope of this nonlinear function correspond to the faces of the so-called cluster weight polytope, for which we give a complete description under certain conditions.<\/jats:p>","DOI":"10.1137\/060675253","type":"journal-article","created":{"date-parts":[[2008,7,23]],"date-time":"2008-07-23T14:45:10Z","timestamp":1216824310000},"page":"1073-1098","source":"Crossref","is-referenced-by-count":5,"title":["On the Graph Bisection Cut Polytope"],"prefix":"10.1137","volume":"22","author":[{"given":"Michael","family":"Armbruster","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Helmberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marzena","family":"F\u00fcgenschuh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexander","family":"Martin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,7,2]]},"reference":[{"key":"R1","unstructured":"M. Armbruster,\n                      Branch-and-Cut for a Semidefinite Relaxation of Large-scale Minimum Bisection Problems\n                      , Ph.D. thesis, Technische Universit\u00e4t Chemnitz, Chemnitz, Germany, 2007."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02592023"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01588778"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01588779"},{"key":"R5","unstructured":"C. C. de Souza,\n                      The Graph Equipartition Problem: Optimal Solutions, Extensions and Applications\n                      , Ph.D. thesis, Universit\u00e9 Catholique de Louvain, Louvain-la-Neuve, Belgium, 1993."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"M. Deza and M. Laurent,\n                      Geometry of Cuts and Metrics\n                      , Algorithms Combin. 15, Springer, Berlin, 1997.","DOI":"10.1007\/978-3-642-04295-9"},{"key":"R7","unstructured":"A. Eisenbl\u00e4tter,\n                      Frequency Assignment in GSM Networks\n                      , Ph.D. thesis, Technische Universit\u00e4t Berlin, Berlin, 2001."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02592198"},{"key":"R9","unstructured":"M. F\u00fcgenschuh,\n                      Relaxations and Solutions for the Minimum Graph Bisection Problem\n                      , Ph.D. thesis, Darmstadt University of Technology, Darmstadt, Germany, 2007."},{"key":"R10","unstructured":"M. R. Garey and D. S. Johnson,\n                      Computers and Intractability\n                      , W. H. Freeman, New York, 1979."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"E. Gawrilow and M. Joswig,\n                      polymake: A framework for analyzing convex polytopes\n                      , in Polytopes\u2014Combinatorics and Computation, G. Kalai and G. M. Ziegler, eds., Birkh\u00e4user, Basel, 2000, pp. 43\u201374.","DOI":"10.1007\/978-3-0348-8438-9_2"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396660"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585164"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(94)00151-3"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"T. Lengauer,\n                      Combinatorial Algorithms for Integrated Circuit Layout\n                      , John Wiley, Chichester, UK, 1990.","DOI":"10.1007\/978-3-322-92106-2"},{"key":"R16","unstructured":"S. Martello and P. Toth,\n                      Knapsack Problems\n                      , John Wiley, Chichester, UK, 1990."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5610(96)00064-0"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060675253","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:05:10Z","timestamp":1787321110000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060675253"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060675253"],"URL":"https:\/\/doi.org\/10.1137\/060675253","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}