An approximation algorithm for the balanced Max-3-Uncut problem using complex semidefinite programming rounding

Graph partition problems have been investigated extensively in combinatorial optimization. In this work, we consider an important graph partition problem which has applications in the design of VLSI circuits, namely, the balanced Max-3-Uncut problem . We formulate the problem as a discrete linear pr...

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Veröffentlicht in:	Journal of combinatorial optimization Jg. 32; H. 4; S. 1017 - 1035
Hauptverfasser:	Wu, Chenchen, Xu, Dachuan, Du, Donglei, Xu, Wenqing
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.11.2016 Springer Nature B.V
Schlagworte:	Algorithms Approximation Bivariate analysis Circuit design Combinatorial analysis Combinatorics Complex variables Convex and Discrete Geometry Integrated circuits Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Partitions (mathematics) Rounding Semidefinite programming Theory of Computation Complex semidefinite programming Approximation algorithm Balanced Max-3-Uncut
ISSN:	1382-6905, 1573-2886
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Abstract	Graph partition problems have been investigated extensively in combinatorial optimization. In this work, we consider an important graph partition problem which has applications in the design of VLSI circuits, namely, the balanced Max-3-Uncut problem . We formulate the problem as a discrete linear program with complex variables and propose an approximation algorithm with an approximation ratio of 0.3456 using a semidefinite programming rounding technique along with a greedy swapping step afterwards to guarantee the balanced constraint. Our analysis utilizes a bivariate function, rather than the univariate function in previous work.
AbstractList	Graph partition problems have been investigated extensively in combinatorial optimization. In this work, we consider an important graph partition problem which has applications in the design of VLSI circuits, namely, the balanced Max-3-Uncut problem . We formulate the problem as a discrete linear program with complex variables and propose an approximation algorithm with an approximation ratio of 0.3456 using a semidefinite programming rounding technique along with a greedy swapping step afterwards to guarantee the balanced constraint. Our analysis utilizes a bivariate function, rather than the univariate function in previous work. Graph partition problems have been investigated extensively in combinatorial optimization. In this work, we consider an important graph partition problem which has applications in the design of VLSI circuits, namely, the balanced Max-3-Uncut problem. We formulate the problem as a discrete linear program with complex variables and propose an approximation algorithm with an approximation ratio of 0.3456 using a semidefinite programming rounding technique along with a greedy swapping step afterwards to guarantee the balanced constraint. Our analysis utilizes a bivariate function, rather than the univariate function in previous work.
Author	Wu, Chenchen Xu, Wenqing Du, Donglei Xu, Dachuan
Author_xml	– sequence: 1 givenname: Chenchen surname: Wu fullname: Wu, Chenchen organization: College of Science, Tianjin University of Technology – sequence: 2 givenname: Dachuan surname: Xu fullname: Xu, Dachuan email: xudc@bjut.edu.cn organization: College of Applied Sciences, Beijing University of Technology – sequence: 3 givenname: Donglei surname: Du fullname: Du, Donglei organization: Faculty of Business Administration, University of New Brunswick – sequence: 4 givenname: Wenqing surname: Xu fullname: Xu, Wenqing organization: College of Applied Sciences, Beijing University of Technology, Department of Mathematics and Statistics, California State University
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References	LasserreJBAn explicit equivalent positive semidefinite program for nonlinear 0–1 programsSIAM J Optim200212756769188491610.1137/S10526234003800791007.90046 GoemansMXWilliamsonDPApproximation algorithms for MAX-3-CUT and other problems via complex semidefinite programmingJ Comput Syst Sci200468442470205910310.1016/j.jcss.2003.07.0121093.90038 Raghavendra P, Tan N (2012) Approximating CSPs with global cardinality constraints using SDP hierarchies. In: 23rd Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM Press, Kyoto, pp 373–387 ZhangSHuangYComplex quadratic optimization and semidefinite programmingSIAM J Optim200616871890219756010.1137/04061341X1113.90115 ChoudhurySGaurDKrishnamurtiRAn approximation algorithm for Max k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Uncut with capacity constraintsOptimization201261143150287932810.1080/02331934.2011.5925271236.05194 XuDHanJHuangZZhangLImproved approximation algorithms for MAX n/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n/2$$\end{document}-DIRECTED-BISECTION and MAX n/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n/2$$\end{document}-DENSE-SUBGRAPHJ Glob Optim200327399410201281310.1023/A:10260941106471046.90094 FeigeULangbergMThe RPR2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm RPR}^{2}$$\end{document} rounding technique for semidefinite programsJ Algorithms200660123222894210.1016/j.jalgor.2004.11.0031113.90116 HanQYeYZhangJAn improved rounding method and semidefinite programming relaxation for graph partitionMath Progr Ser B200292509535190576410.1007/s1010701002881008.90042 Austrin P, Benabbas S, Georgiou K (2013) Better balance by being biased: A 0.8776-approximation for Max Bisection. 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References_xml	– reference: LasserreJBAn explicit equivalent positive semidefinite program for nonlinear 0–1 programsSIAM J Optim200212756769188491610.1137/S10526234003800791007.90046 – reference: Zwick U (1999) Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: 31st Annual ACM Symposium on Theory of Computing. ACM Press, Atlanta, pp 679–687 – reference: FriezeAMJerrumMImproved approximation algorithms for MAX k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-CUT and MAX BISECTIONAlgorithmica1997186781143202910.1007/BF025236880873.68078 – reference: GoemansMXWilliamsonDPApproximation algorithms for MAX-3-CUT and other problems via complex semidefinite programmingJ Comput Syst Sci200468442470205910310.1016/j.jcss.2003.07.0121093.90038 – reference: Andersson G (1999) An approximation algorithm for Max p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-Section. In: 16th Annual Symposium on Theoretical Aspects of Computer Science. Springer Press, Trier, pp 237–247 – reference: XuDHanJHuangZZhangLImproved approximation algorithms for MAX n/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n/2$$\end{document}-DIRECTED-BISECTION and MAX n/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n/2$$\end{document}-DENSE-SUBGRAPHJ Glob Optim200327399410201281310.1023/A:10260941106471046.90094 – reference: WuCDuDXuDAn improved semidefinite programming hierarchies rounding approximation algorithm for maximum graph bisection problemsJ Comb Optim2015295366329625610.1007/s10878-013-9673-11327.90183 – reference: YeYA .699-approximation algorithm for Max-BisectionMath Program200190101111181978810.1007/PL000114151059.90119 – reference: KleinbergJPapadimitriouCRaghavanPSegmentation problemsJ ACM200451116214565510.1145/972639.9726441317.90329 – reference: Austrin P, Benabbas S, Georgiou K (2013) Better balance by being biased: A 0.8776-approximation for Max Bisection. In: 24th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM Press, New Orleans, pp 277–294 – reference: FeoTGoldschmidtOKhellafMOne-half approximation algorithms for the k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-partition problemOper Res199240S170S17310.1287/opre.40.1.S1700745.90072 – reference: GoemansMXWilliamsonDPImproved approximation algorithms for maximum cut and satisfiability problems using semidefinite programmingJ ACM19954211151145141222810.1145/227683.2276840885.68088 – reference: SahniSGonzalezTP-complete approximation problemsJ ACM19762355556540831310.1145/321958.3219750348.90152 – reference: ZhangSHuangYComplex quadratic optimization and semidefinite programmingSIAM J Optim200616871890219756010.1137/04061341X1113.90115 – reference: HalperinEZwickUA unified framework for obtaining improved approximation algorithms for maximum graph bisection problemsRandom Struct Algorithms200220382402190061410.1002/rsa.100351017.68089 – reference: ChoudhurySGaurDKrishnamurtiRAn approximation algorithm for Max k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Uncut with capacity constraintsOptimization201261143150287932810.1080/02331934.2011.5925271236.05194 – reference: Ling A (2009) Approximation algorithms for Max 3-Section using complex semidefinite programming relaxation. In: 3rd International Conference on Combinatorial Optimization and Applications. Springer Press, Huangshan, pp 219–230 – reference: FeigeULangbergMThe RPR2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm RPR}^{2}$$\end{document} rounding technique for semidefinite programsJ Algorithms200660123222894210.1016/j.jalgor.2004.11.0031113.90116 – reference: Doids Y, Guruswami V, Khanna S (1999) The 2-catalog segmentation problem. In: 17th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM Press, Baltimore, pp 897–898 – reference: HanQYeYZhangJAn improved rounding method and semidefinite programming relaxation for graph partitionMath Progr Ser B200292509535190576410.1007/s1010701002881008.90042 – reference: Raghavendra P, Tan N (2012) Approximating CSPs with global cardinality constraints using SDP hierarchies. In: 23rd Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM Press, Kyoto, pp 373–387 – volume: 61 start-page: 143 year: 2012 ident: 9880_CR3 publication-title: Optimization doi: 10.1080/02331934.2011.592527 – volume: 29 start-page: 53 year: 2015 ident: 9880_CR17 publication-title: J Comb Optim doi: 10.1007/s10878-013-9673-1 – ident: 9880_CR15 doi: 10.1137/1.9781611973099.33 – volume: 16 start-page: 871 year: 2006 ident: 9880_CR20 publication-title: SIAM J Optim doi: 10.1137/04061341X – ident: 9880_CR2 doi: 10.1137/1.9781611973105.21 – volume: 18 start-page: 67 year: 1997 ident: 9880_CR7 publication-title: Algorithmica doi: 10.1007/BF02523688 – volume: 51 start-page: 1 year: 2004 ident: 9880_CR12 publication-title: J ACM doi: 10.1145/972639.972644 – ident: 9880_CR4 – volume: 42 start-page: 1115 year: 1995 ident: 9880_CR8 publication-title: J ACM doi: 10.1145/227683.227684 – ident: 9880_CR14 doi: 10.1007/978-3-642-02026-1_20 – volume: 20 start-page: 382 year: 2002 ident: 9880_CR10 publication-title: Random Struct Algorithms doi: 10.1002/rsa.10035 – volume: 90 start-page: 101 year: 2001 ident: 9880_CR19 publication-title: Math Program doi: 10.1007/PL00011415 – volume: 23 start-page: 555 year: 1976 ident: 9880_CR16 publication-title: J ACM doi: 10.1145/321958.321975 – volume: 60 start-page: 1 year: 2006 ident: 9880_CR5 publication-title: J Algorithms doi: 10.1016/j.jalgor.2004.11.003 – volume: 40 start-page: S170 year: 1992 ident: 9880_CR6 publication-title: Oper Res doi: 10.1287/opre.40.1.S170 – ident: 9880_CR21 doi: 10.1145/301250.301431 – ident: 9880_CR1 doi: 10.1007/3-540-49116-3_22 – volume: 92 start-page: 509 year: 2002 ident: 9880_CR11 publication-title: Math Progr Ser B doi: 10.1007/s101070100288 – volume: 27 start-page: 399 year: 2003 ident: 9880_CR18 publication-title: J Glob Optim doi: 10.1023/A:1026094110647 – volume: 68 start-page: 442 year: 2004 ident: 9880_CR9 publication-title: J Comput Syst Sci doi: 10.1016/j.jcss.2003.07.012 – volume: 12 start-page: 756 year: 2002 ident: 9880_CR13 publication-title: SIAM J Optim doi: 10.1137/S1052623400380079
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SubjectTerms	Algorithms Approximation Bivariate analysis Circuit design Combinatorial analysis Combinatorics Complex variables Convex and Discrete Geometry Integrated circuits Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Partitions (mathematics) Rounding Semidefinite programming Theory of Computation
Title	An approximation algorithm for the balanced Max-3-Uncut problem using complex semidefinite programming rounding
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