A maximum hypergraph 3-cut problem with limited unbalance: approximation and analysis
We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph H = ( V , E , w ) into three disjoint subsets V 1 , V 2 , and V 3 such that the sum of edge weights cross different parts is maximized subject to |...
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| Published in: | Journal of global optimization Vol. 87; no. 2-4; pp. 917 - 937 |
|---|---|
| Main Authors: | , , , , , |
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| Language: | English |
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01.11.2023
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| ISSN: | 0925-5001, 1573-2916 |
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| Abstract | We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph
H
=
(
V
,
E
,
w
)
into three disjoint subsets
V
1
,
V
2
, and
V
3
such that the sum of edge weights cross different parts is maximized subject to
|
|
V
i
|
-
|
V
l
|
|
≤
B
(
∀
i
≠
l
∈
{
1
,
2
,
3
}
) for a given parameter
B
. This problem is NP-hard because it includes some well-known problems like the max 3-section problem and the max 3-cut problem as special cases. We formulate the MH3C-LU as a ternary quadratic program and present a randomized approximation algorithm based on the complex semidefinite programming relaxation technique. |
|---|---|
| AbstractList | We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph
H
=
(
V
,
E
,
w
)
into three disjoint subsets
V
1
,
V
2
, and
V
3
such that the sum of edge weights cross different parts is maximized subject to
|
|
V
i
|
-
|
V
l
|
|
≤
B
(
∀
i
≠
l
∈
{
1
,
2
,
3
}
) for a given parameter
B
. This problem is NP-hard because it includes some well-known problems like the max 3-section problem and the max 3-cut problem as special cases. We formulate the MH3C-LU as a ternary quadratic program and present a randomized approximation algorithm based on the complex semidefinite programming relaxation technique. We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph [Formula omitted] into three disjoint subsets [Formula omitted], [Formula omitted], and [Formula omitted] such that the sum of edge weights cross different parts is maximized subject to [Formula omitted] ( [Formula omitted]) for a given parameter B. This problem is NP-hard because it includes some well-known problems like the max 3-section problem and the max 3-cut problem as special cases. We formulate the MH3C-LU as a ternary quadratic program and present a randomized approximation algorithm based on the complex semidefinite programming relaxation technique. |
| Audience | Academic |
| Author | Zhang, Zan-Bo Chen, Yannan Han, Deren Sun, Jian Du, Donglei Zhang, Xiaoyan |
| Author_xml | – sequence: 1 givenname: Jian surname: Sun fullname: Sun, Jian organization: Department of Operations Research and Information Engineering, Beijing University of Technology – sequence: 2 givenname: Zan-Bo surname: Zhang fullname: Zhang, Zan-Bo organization: School of Statistics and Mathematics, Guangdong University of Finance and Economics – sequence: 3 givenname: Yannan surname: Chen fullname: Chen, Yannan organization: School of Mathematical Sciences, South China Normal University – sequence: 4 givenname: Deren surname: Han fullname: Han, Deren organization: School of Mathematical Sciences, Beijing Advanced Innovation Center for Big Data and Brain Computing (BDBC), Beihang University – sequence: 5 givenname: Donglei surname: Du fullname: Du, Donglei organization: Faculty of Management, University of New Brunswick – sequence: 6 givenname: Xiaoyan orcidid: 0000-0002-2224-1484 surname: Zhang fullname: Zhang, Xiaoyan email: zhangxiaoyan@njnu.edu.cn organization: School of Mathematical Science and Institute of Mathematics, Nanjing Normal University |
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| Issue | 2-4 |
| Keywords | Complex semidefinite programming Randomized algorithm Approximation algorithm Max hypergraph 3-cut |
| Language | English |
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| PublicationSubtitle | An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering |
| PublicationTitle | Journal of global optimization |
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| References | GoemansM-XWilliamsonD-PApproximation algorithms for max-3-cut and other problems via complex semidefinite programmingJ. Comput. Syst. Sci.2004682442470205910310.1016/j.jcss.2003.07.0121093.90038 XuZHongMLuoZ-QSemidefinite approximation for mixed binary quadratically constrained quadratic programsSIAM J. Optim.201424312651293324804110.1137/1309095971321.90101 BarahonaFGrötschelMReineltGAn application of combinatorial optimization to statistical physics and circuit layout designOper. Res.198836349351310.1287/opre.36.3.4930646.90084 HuangYZhangSApproximation algorithms for indefinite complex quadratic maximization problemsSci. China Math.2010531026972708272827210.1007/s11425-010-3087-71209.90284 LuCLiuY-FZhangW-QZhangS-ZTightness of a new and enhanced semidefinite relaxation for MIMO detectionSIAM J. Optim.2019291719742391941410.1137/17M115075X1412.90104 Hayrapetyan, A., Kempe, D., Pal, M., Svitkina, Z.: Unbalance graph cuts. In: Proceedings of the 13th Annual European Symposium, pp. 191–202 (2005) SahniSGonzalesTP-complete approximation problemsJ. ACM197623355556540831310.1145/321958.3219750348.90152 GoemansM-XWilliamsonD-PImproved approximation algorithms for maximum Cut and satisfiability problem using semidefinite programmingJ. ACM199542611151145141222810.1145/227683.2276840885.68088 AnderssonGEngebretsenLBetter approximation algorithms for set splitting and NOT-ALL-EQUAL SATInf. Process. Lett.1998656305311162165110.1016/S0020-0190(98)00021-01338.68286 XiaYNew semidefinite programming relaxations for box constrained quadratic programSci. China Math.2013564877886303484810.1007/s11425-012-4512-x1302.90135 XuBYuXZhangXZhangZAn SDP randomized approximation algorithm for max hypergraph cut with limited unbalanceSci. China Math.2014571224372462327539610.1007/s11425-014-4900-51335.49053 FriezeA-MJerrumMImproved 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 bisectionAlgorithmica19971816781143202910.1007/BF025236880873.68078 HalperinEZwickUA unified framework for obtaining improved approximation algorithms for maximum graph bisection problemsRandom Struct. Algorithms2002203382402190061410.1002/rsa.100351017.68089 Raghavendra, P., Tan, N.: Approximating CSPs with global cardinality constraints using SDP hierarchies. In: Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, pp. 373–387 (2012) ZhangJYeYHanQImproved approximations for max set splitting and max NAE SATDiscrete Appl. Math.20041421–3133149207508810.1016/j.dam.2002.07.0011122.68154 Andersson, G.: 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: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, vol. 1563, pp. 237–247 (1999) KhotSKindlerGMosselEO’DonnellROptimal inapproximability results formax-cut and other 2-variable CSPs?SIAM J. Comput.2007371319357230629510.1137/S00975397054473721135.68019 FeigeULangbergMThe RPR2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{2}$$\end{document} rounding technique for semidefinite programsJ. Algorithms2006601123222894210.1016/j.jalgor.2004.11.0031113.90116 GalbiatiGMaffioliFApproximation algorithms for maximum cut with limited unbalanceTheoret. Comput. Sci.20073851–37887235624310.1016/j.tcs.2007.05.0361124.68117 AustrinPBenabbasSGeorgiouKBetter balance by being biased: a 0.8776-approximation for max bisectionACM Trans. Algorithms2016131127359810610.1145/29070521422.68283 Ageev, A.-A., Sviridenko, M.-I.: Approximation algorithms for maximum coverage and Max-Cut with given sizes of parts. In: Proceedings of the 7th Integer Programming and Combinatorial Optimization, pp. 17–30 (1999) YeYA. 699-approximation algorithm for max-bisectionMath. Program.2001901101111181978810.1007/PL000114151059.90119 G Andersson (1183_CR3) 1998; 65 S Khot (1183_CR14) 2007; 37 C Lu (1183_CR15) 2019; 29 G Galbiati (1183_CR8) 2007; 385 S Sahni (1183_CR17) 1976; 23 1183_CR12 Y Xia (1183_CR18) 2013; 56 1183_CR16 1183_CR2 P Austrin (1183_CR4) 2016; 13 1183_CR1 E Halperin (1183_CR11) 2002; 20 Y Huang (1183_CR13) 2010; 53 M-X Goemans (1183_CR9) 1995; 42 F Barahona (1183_CR5) 1988; 36 Z Xu (1183_CR20) 2014; 24 B Xu (1183_CR19) 2014; 57 Y Ye (1183_CR21) 2001; 90 M-X Goemans (1183_CR10) 2004; 68 U Feige (1183_CR6) 2006; 60 A-M Frieze (1183_CR7) 1997; 18 J Zhang (1183_CR22) 2004; 142 |
| References_xml | – reference: Andersson, G.: 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: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, vol. 1563, pp. 237–247 (1999) – reference: GoemansM-XWilliamsonD-PImproved approximation algorithms for maximum Cut and satisfiability problem using semidefinite programmingJ. ACM199542611151145141222810.1145/227683.2276840885.68088 – reference: KhotSKindlerGMosselEO’DonnellROptimal inapproximability results formax-cut and other 2-variable CSPs?SIAM J. Comput.2007371319357230629510.1137/S00975397054473721135.68019 – reference: XuBYuXZhangXZhangZAn SDP randomized approximation algorithm for max hypergraph cut with limited unbalanceSci. China Math.2014571224372462327539610.1007/s11425-014-4900-51335.49053 – reference: YeYA. 699-approximation algorithm for max-bisectionMath. Program.2001901101111181978810.1007/PL000114151059.90119 – reference: HalperinEZwickUA unified framework for obtaining improved approximation algorithms for maximum graph bisection problemsRandom Struct. Algorithms2002203382402190061410.1002/rsa.100351017.68089 – reference: LuCLiuY-FZhangW-QZhangS-ZTightness of a new and enhanced semidefinite relaxation for MIMO detectionSIAM J. Optim.2019291719742391941410.1137/17M115075X1412.90104 – reference: Raghavendra, P., Tan, N.: Approximating CSPs with global cardinality constraints using SDP hierarchies. In: Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, pp. 373–387 (2012) – reference: Hayrapetyan, A., Kempe, D., Pal, M., Svitkina, Z.: Unbalance graph cuts. In: Proceedings of the 13th Annual European Symposium, pp. 191–202 (2005) – reference: HuangYZhangSApproximation algorithms for indefinite complex quadratic maximization problemsSci. China Math.2010531026972708272827210.1007/s11425-010-3087-71209.90284 – 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}$$^{2}$$\end{document} rounding technique for semidefinite programsJ. Algorithms2006601123222894210.1016/j.jalgor.2004.11.0031113.90116 – reference: GoemansM-XWilliamsonD-PApproximation algorithms for max-3-cut and other problems via complex semidefinite programmingJ. Comput. Syst. Sci.2004682442470205910310.1016/j.jcss.2003.07.0121093.90038 – reference: XuZHongMLuoZ-QSemidefinite approximation for mixed binary quadratically constrained quadratic programsSIAM J. Optim.201424312651293324804110.1137/1309095971321.90101 – reference: XiaYNew semidefinite programming relaxations for box constrained quadratic programSci. China Math.2013564877886303484810.1007/s11425-012-4512-x1302.90135 – reference: ZhangJYeYHanQImproved approximations for max set splitting and max NAE SATDiscrete Appl. Math.20041421–3133149207508810.1016/j.dam.2002.07.0011122.68154 – reference: AustrinPBenabbasSGeorgiouKBetter balance by being biased: a 0.8776-approximation for max bisectionACM Trans. Algorithms2016131127359810610.1145/29070521422.68283 – reference: FriezeA-MJerrumMImproved 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 bisectionAlgorithmica19971816781143202910.1007/BF025236880873.68078 – reference: BarahonaFGrötschelMReineltGAn application of combinatorial optimization to statistical physics and circuit layout designOper. Res.198836349351310.1287/opre.36.3.4930646.90084 – reference: AnderssonGEngebretsenLBetter approximation algorithms for set splitting and NOT-ALL-EQUAL SATInf. Process. Lett.1998656305311162165110.1016/S0020-0190(98)00021-01338.68286 – reference: SahniSGonzalesTP-complete approximation problemsJ. 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| Snippet | We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph
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V... We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph... |
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| SubjectTerms | Algorithms Analysis Computer Science Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions Relaxation |
| Title | A maximum hypergraph 3-cut problem with limited unbalance: approximation and analysis |
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