A polynomial time algorithm for finding a minimum 4-partition of a submodular function

In this paper, we study the minimum k -partition problem of submodular functions, i.e., given a finite set V and a submodular function f : 2 V → R , computing a k -partition { V 1 , … , V k } of V with minimum ∑ i = 1 k f ( V i ) . The problem is a natural generalization of the minimum k -cut proble...

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Veröffentlicht in:	Mathematical programming Jg. 207; H. 1-2; S. 717 - 732
Hauptverfasser:	Hirayama, Tsuyoshi, Liu, Yuhao, Makino, Kazuhisa, Shi, Ke, Xu, Chao
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2024 Springer
Schlagworte:	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical Submodular function 90C27 Combinatorial optimization Polynomial time
ISSN:	0025-5610, 1436-4646
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Abstract	In this paper, we study the minimum k -partition problem of submodular functions, i.e., given a finite set V and a submodular function f : 2 V → R , computing a k -partition { V 1 , … , V k } of V with minimum ∑ i = 1 k f ( V i ) . The problem is a natural generalization of the minimum k -cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k , and solvable in polynomial time for fixed k ≤ 3 . In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem.
AbstractList	In this paper, we study the minimum k-partition problem of submodular functions, i.e., given a finite set V and a submodular function [Formula omitted], computing a k-partition [Formula omitted] of V with minimum [Formula omitted]. The problem is a natural generalization of the minimum k-cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k, and solvable in polynomial time for fixed [Formula omitted]. In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem. In this paper, we study the minimum k -partition problem of submodular functions, i.e., given a finite set V and a submodular function f : 2 V → R , computing a k -partition { V 1 , … , V k } of V with minimum ∑ i = 1 k f ( V i ) . The problem is a natural generalization of the minimum k -cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k , and solvable in polynomial time for fixed k ≤ 3 . In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem.
Audience	Academic
Author	Hirayama, Tsuyoshi Makino, Kazuhisa Shi, Ke Liu, Yuhao Xu, Chao
Author_xml	– sequence: 1 givenname: Tsuyoshi surname: Hirayama fullname: Hirayama, Tsuyoshi organization: Toshiba Digital Solutions Corporation – sequence: 2 givenname: Yuhao surname: Liu fullname: Liu, Yuhao organization: University of Electronic Science and Technology of China – sequence: 3 givenname: Kazuhisa surname: Makino fullname: Makino, Kazuhisa organization: Kyoto University – sequence: 4 givenname: Ke surname: Shi fullname: Shi, Ke organization: University of Electronic Science and Technology of China – sequence: 5 givenname: Chao orcidid: 0000-0003-4417-3299 surname: Xu fullname: Xu, Chao email: the.chao.xu@gmail.com organization: University of Electronic Science and Technology of China
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References	Gupta, A., Lee, E., Li, J.: Faster exact and approximate algorithms for k-cut. In: 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 113–123 (2018). https://doi.org/10.1109/FOCS.2018.00020 Beideman, C., Chandrasekaran, K., Wang, W.: Deterministic enumeration of all minimum 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-sets in hypergraphs for fixed 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}. In: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2208–2228. Society for Industrial and Applied Mathematics, Philadelphia (2022). https://doi.org/10.1137/1.9781611977073 Hirayama, T., Liu, Y., Makino, K., Shi, K., Xu, C.: A polynomial time algorithm for finding a minimum 4-partition of a submodular function. In: Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1680–1691. https://doi.org/10.1137/1.9781611977554.ch64 Fox, K., Panigrahi, D., Zhang, F.: Minimum cut and minimum 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 in hypergraphs via branching contractions. In: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 881–896. SIAM (2019) Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Tech. Rep. B 96-02, FU Berlin Fachbereich Mathematik und Informatik (1996) Guinez, F., Queyranne, M.: The size-constrained submodular 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 problem (2012). https://docs.google.com/viewer?a=v &pid=sites &srcid=ZGVmYXVsdGRvbWFpbnxmbGF2aW9ndWluZXpob21lcGFnZXxneDo0NDVlMThkMDg4ZWRlOGI1. Unpublished manuscript. See also https://smartech.gatech.edu/bitstream/handle/1853/43309/Queyranne.pdf GoldschmidtOHochbaumDSA polynomial algorithm 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}-cut problem for fixed 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}Math. Oper. Res.1994192437129000810.1287/moor.19.1.24 NagamochiHAlgorithms for the minimum partitioning problems in graphsElectron. Commun. Jpn.20079010637810.1002/ecjc.20341Part III Fundamental Electronic Science Vazirani, V.V., Yannakakis, M.: Suboptimal cuts: Their enumeration, weight and number. In: International Colloquium on Automata, Languages, and Programming, pp. 366–377. Springer (1992) GasieniecLJanssonJLingasAÖstlinAOn the complexity of constructing evolutionary treesJ. Comb. Optim.199932/3183197170861610.1023/A:1009833626004 NagamochiHIbarakiTComputing edge-connectivity in multigraphs and capacitated graphsSIAM J. Discrete Math.1992515466114689610.1137/0405004 ChandrasekaranKXuCYuXHypergraph 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 in randomized polynomial timeMath. Program.20211861–285113421447710.1007/s10107-019-01443-7 FukunagaTComputing minimum multiway cuts in hypergraphsDiscrete Optim.2013104371382312862410.1016/j.disopt.2013.10.002 XiaoMHongSHNagamochiHFukunagaTAn improved divide-and-conquer algorithm for finding all minimum k-way cutsAlgorithms and Computation2008BerlinSpringer20821910.1007/978-3-540-92182-0_21 BeidemanCChandrasekaranKXuCMulticriteria cuts and size-constrained 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}-cuts in hypergraphsMath. Program.202110.1007/s10107-021-01732-0 OkumotoKFukunagaTNagamochiHDivide-and-conquer algorithms for partitioning hypergraphs and submodular systemsAlgorithmica201010.1007/s00453-010-9483-0 ZhaoLNagamochiHIbarakiTOn generalized greedy splitting algorithms for multiway partition problemsDiscrete Appl. Math.20041431130143208787510.1016/j.dam.2003.10.007 GuptaAHarrisDGLeeELiJOptimal bounds 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}-cut problemJ. ACM202110.1145/3478018 NagamochiHKatayamaSIbarakiTA faster algorithm for computing minimum 5-way and 6-way cuts in graphsJ. Comb. Optim.200042151169177282310.1023/A:1009804919645 ChandrasekaranKChekuriCHypergraph 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 for fixed 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} in deterministic polynomial timeMath. Oper. Res.2022451550710.1287/moor.2021.1250 NagamochiHIbarakiTA fast algorithm for computing minimum 3-way and 4-way cutsMath. Program.2000883507520178215310.1007/PL00011383 ChekuriCXuCMinimum cuts and sparsification in hypergraphsSIAM J. Comput.201847621182156388124610.1137/18M1163865 XiaoMFinding minimum 3-way cuts in hypergraphsInf. Process. Lett.201011014–15554558267497010.1016/j.ipl.2010.05.003 Chekuri, C., Ene, A.: Approximation algorithms for submodular multiway partition. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 807–816. IEEE, Palm Springs, CA, USA (2011). https://doi.org/10.1109/FOCS.2011.34 OkumotoKFukunagaTNagamochiHDongYDuDZIbarraODivide-and-conquer algorithms for partitioning hypergraphs and submodular systemsAlgorithms and Computation2009BerlinSpringer556410.1007/978-3-642-10631-6_8 Chandrasekaran, K., Chekuri, C.: Min–max partitioning of hypergraphs and symmetric submodular functions. In: Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’21, pp. 1026–1038. Society for Industrial and Applied Mathematics, USA (2021) QueyranneMMinimizing symmetric submodular functionsMath. Program.1998821312162769310.1007/BF01585863 YehLPWangBFSuHHEfficient algorithms for the problems of enumerating cuts by non-decreasing weightsAlgorithmica2010563297312258106810.1007/s00453-009-9284-5 Lee, Y.T., Sidford, A., Wong, S.C.W.: A faster cutting plane method and its implications for combinatorial and convex optimization. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pp. 1049–1065 (2015). https://doi.org/10.1109/FOCS.2015.68 KamidoiYYoshidaNNagamochiHA deterministic algorithm for finding all minimum 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}-way cutsSIAM J. Comput.200636513291341228408310.1137/050631616 Thorup, M.: Minimum 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}-way cuts via deterministic greedy tree packing. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 159–165 (2008). https://doi.org/10.1145/1374376.1374402 Beideman, C., Chandrasekaran, K., Wang, W.: Counting and enumerating optimum cut sets for hypergraph 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}-partitioning problems for fixed k. In: Bojańczyk, M., Merelli, E., Woodruff, D.P. (eds.) 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), Leibniz International Proceedings in Informatics (LIPIcs), vol. 229, pp. 16:1–16:18. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022). https://doi.org/10.4230/LIPIcs.ICALP.2022.16. https://drops.dagstuhl.de/opus/volltexte/2022/16357 MakWKWongDFFast hypergraph min-cut algorithm for circuit partitioningIntegr. VLSI J.200030111110.1016/S0167-9260(00)00008-0 ZhaoLNagamochiHIbarakiTGreedy splitting algorithms for approximating multiway partition problemsMath. Program.20051021167183211548510.1007/s10107-004-0510-2 ChekuriCQuanrudKXuCLP Relaxation and tree packing for minimum 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 O Goldschmidt (2029_CR13) 1994; 19 M Queyranne (2029_CR28) 1998; 82 M Xiao (2029_CR32) 2010; 110 T Fukunaga (2029_CR11) 2013; 10 L Gasieniec (2029_CR12) 1999; 3 2029_CR20 L Zhao (2029_CR34) 2004; 143 K Okumoto (2029_CR27) 2010 K Chandrasekaran (2029_CR5) 2022 H Nagamochi (2029_CR24) 2000; 88 2029_CR29 2029_CR14 2029_CR16 Y Kamidoi (2029_CR18) 2006; 36 2029_CR17 2029_CR10 K Okumoto (2029_CR26) 2009 LP Yeh (2029_CR33) 2010; 56 2029_CR7 2029_CR30 2029_CR4 2029_CR2 C Chekuri (2029_CR9) 2018; 47 H Nagamochi (2029_CR22) 2007; 90 M Xiao (2029_CR31) 2008 2029_CR1 WK Mak (2029_CR21) 2000; 30 L Zhao (2029_CR35) 2005; 102 C Beideman (2029_CR3) 2021 K Chandrasekaran (2029_CR6) 2021; 186 H Nagamochi (2029_CR25) 2000; 4 C Chekuri (2029_CR8) 2020; 34 A Gupta (2029_CR15) 2021 2029_CR19 H Nagamochi (2029_CR23) 1992; 5
References_xml	– reference: Vazirani, V.V., Yannakakis, M.: Suboptimal cuts: Their enumeration, weight and number. In: International Colloquium on Automata, Languages, and Programming, pp. 366–377. Springer (1992) – reference: Hirayama, T., Liu, Y., Makino, K., Shi, K., Xu, C.: A polynomial time algorithm for finding a minimum 4-partition of a submodular function. In: Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1680–1691. https://doi.org/10.1137/1.9781611977554.ch64 – reference: Chekuri, C., Ene, A.: Approximation algorithms for submodular multiway partition. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 807–816. IEEE, Palm Springs, CA, USA (2011). https://doi.org/10.1109/FOCS.2011.34 – reference: OkumotoKFukunagaTNagamochiHDongYDuDZIbarraODivide-and-conquer algorithms for partitioning hypergraphs and submodular systemsAlgorithms and Computation2009BerlinSpringer556410.1007/978-3-642-10631-6_8 – reference: Beideman, C., Chandrasekaran, K., Wang, W.: Counting and enumerating optimum cut sets for hypergraph 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}-partitioning problems for fixed k. In: Bojańczyk, M., Merelli, E., Woodruff, D.P. (eds.) 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), Leibniz International Proceedings in Informatics (LIPIcs), vol. 229, pp. 16:1–16:18. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022). https://doi.org/10.4230/LIPIcs.ICALP.2022.16. https://drops.dagstuhl.de/opus/volltexte/2022/16357 – reference: FukunagaTComputing minimum multiway cuts in hypergraphsDiscrete Optim.2013104371382312862410.1016/j.disopt.2013.10.002 – reference: NagamochiHIbarakiTA fast algorithm for computing minimum 3-way and 4-way cutsMath. Program.2000883507520178215310.1007/PL00011383 – reference: GasieniecLJanssonJLingasAÖstlinAOn the complexity of constructing evolutionary treesJ. Comb. Optim.199932/3183197170861610.1023/A:1009833626004 – reference: KamidoiYYoshidaNNagamochiHA deterministic algorithm for finding all minimum 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}-way cutsSIAM J. Comput.200636513291341228408310.1137/050631616 – reference: ChandrasekaranKChekuriCHypergraph 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 for fixed 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} in deterministic polynomial timeMath. Oper. Res.2022451550710.1287/moor.2021.1250 – reference: ChekuriCXuCMinimum cuts and sparsification in hypergraphsSIAM J. Comput.201847621182156388124610.1137/18M1163865 – reference: MakWKWongDFFast hypergraph min-cut algorithm for circuit partitioningIntegr. VLSI J.200030111110.1016/S0167-9260(00)00008-0 – reference: Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Tech. Rep. B 96-02, FU Berlin Fachbereich Mathematik und Informatik (1996) – reference: Guinez, F., Queyranne, M.: The size-constrained submodular 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 problem (2012). https://docs.google.com/viewer?a=v &pid=sites &srcid=ZGVmYXVsdGRvbWFpbnxmbGF2aW9ndWluZXpob21lcGFnZXxneDo0NDVlMThkMDg4ZWRlOGI1. Unpublished manuscript. See also https://smartech.gatech.edu/bitstream/handle/1853/43309/Queyranne.pdf – reference: Chandrasekaran, K., Chekuri, C.: Min–max partitioning of hypergraphs and symmetric submodular functions. In: Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’21, pp. 1026–1038. Society for Industrial and Applied Mathematics, USA (2021) – reference: Lee, Y.T., Sidford, A., Wong, S.C.W.: A faster cutting plane method and its implications for combinatorial and convex optimization. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pp. 1049–1065 (2015). https://doi.org/10.1109/FOCS.2015.68 – reference: Beideman, C., Chandrasekaran, K., Wang, W.: Deterministic enumeration of all minimum 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-sets in hypergraphs for fixed 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}. In: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2208–2228. Society for Industrial and Applied Mathematics, Philadelphia (2022). https://doi.org/10.1137/1.9781611977073 – reference: ChekuriCQuanrudKXuCLP Relaxation and tree packing for minimum 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}-cutSIAM J. Discrete Math.202034213341353411166810.1137/19M1299359 – reference: Gupta, A., Lee, E., Li, J.: Faster exact and approximate algorithms for k-cut. In: 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 113–123 (2018). https://doi.org/10.1109/FOCS.2018.00020 – reference: NagamochiHKatayamaSIbarakiTA faster algorithm for computing minimum 5-way and 6-way cuts in graphsJ. Comb. Optim.200042151169177282310.1023/A:1009804919645 – reference: GoldschmidtOHochbaumDSA polynomial algorithm 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}-cut problem for fixed 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}Math. Oper. Res.1994192437129000810.1287/moor.19.1.24 – reference: YehLPWangBFSuHHEfficient algorithms for the problems of enumerating cuts by non-decreasing weightsAlgorithmica2010563297312258106810.1007/s00453-009-9284-5 – reference: GuptaAHarrisDGLeeELiJOptimal bounds 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}-cut problemJ. ACM202110.1145/3478018 – reference: Fox, K., Panigrahi, D., Zhang, F.: Minimum cut and minimum 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 in hypergraphs via branching contractions. In: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 881–896. SIAM (2019) – reference: ZhaoLNagamochiHIbarakiTGreedy splitting algorithms for approximating multiway partition problemsMath. Program.20051021167183211548510.1007/s10107-004-0510-2 – reference: ZhaoLNagamochiHIbarakiTOn generalized greedy splitting algorithms for multiway partition problemsDiscrete Appl. Math.20041431130143208787510.1016/j.dam.2003.10.007 – reference: OkumotoKFukunagaTNagamochiHDivide-and-conquer algorithms for partitioning hypergraphs and submodular systemsAlgorithmica201010.1007/s00453-010-9483-0 – reference: Thorup, M.: Minimum 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}-way cuts via deterministic greedy tree packing. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 159–165 (2008). https://doi.org/10.1145/1374376.1374402 – reference: XiaoMHongSHNagamochiHFukunagaTAn improved divide-and-conquer algorithm for finding all minimum k-way cutsAlgorithms and Computation2008BerlinSpringer20821910.1007/978-3-540-92182-0_21 – reference: QueyranneMMinimizing symmetric submodular functionsMath. Program.1998821312162769310.1007/BF01585863 – reference: NagamochiHIbarakiTComputing edge-connectivity in multigraphs and capacitated graphsSIAM J. Discrete Math.1992515466114689610.1137/0405004 – reference: BeidemanCChandrasekaranKXuCMulticriteria cuts and size-constrained 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}-cuts in hypergraphsMath. Program.202110.1007/s10107-021-01732-0 – reference: ChandrasekaranKXuCYuXHypergraph 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 in randomized polynomial timeMath. Program.20211861–285113421447710.1007/s10107-019-01443-7 – reference: XiaoMFinding minimum 3-way cuts in hypergraphsInf. Process. Lett.201011014–15554558267497010.1016/j.ipl.2010.05.003 – reference: NagamochiHAlgorithms for the minimum partitioning problems in graphsElectron. Commun. Jpn.20079010637810.1002/ecjc.20341Part III Fundamental Electronic Science – volume: 110 start-page: 554 issue: 14–15 year: 2010 ident: 2029_CR32 publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2010.05.003 – volume: 19 start-page: 24 year: 1994 ident: 2029_CR13 publication-title: Math. Oper. Res. doi: 10.1287/moor.19.1.24 – volume: 4 start-page: 151 issue: 2 year: 2000 ident: 2029_CR25 publication-title: J. Comb. Optim. doi: 10.1023/A:1009804919645 – ident: 2029_CR4 doi: 10.1137/1.9781611976465.64 – ident: 2029_CR20 doi: 10.1109/FOCS.2015.68 – ident: 2029_CR30 doi: 10.1007/3-540-55719-9_88 – ident: 2029_CR17 doi: 10.1137/1.9781611977554.ch64 – volume: 5 start-page: 54 issue: 1 year: 1992 ident: 2029_CR23 publication-title: SIAM J. Discrete Math. doi: 10.1137/0405004 – start-page: 55 volume-title: Algorithms and Computation year: 2009 ident: 2029_CR26 doi: 10.1007/978-3-642-10631-6_8 – ident: 2029_CR1 doi: 10.4230/LIPIcs.ICALP.2022.16 – volume: 186 start-page: 85 issue: 1–2 year: 2021 ident: 2029_CR6 publication-title: Math. Program. doi: 10.1007/s10107-019-01443-7 – year: 2021 ident: 2029_CR15 publication-title: J. ACM doi: 10.1145/3478018 – volume: 143 start-page: 130 issue: 1 year: 2004 ident: 2029_CR34 publication-title: Discrete Appl. Math. doi: 10.1016/j.dam.2003.10.007 – year: 2010 ident: 2029_CR27 publication-title: Algorithmica doi: 10.1007/s00453-010-9483-0 – ident: 2029_CR10 doi: 10.1137/1.9781611975482.54 – ident: 2029_CR14 – ident: 2029_CR7 doi: 10.1109/FOCS.2011.34 – volume: 34 start-page: 1334 issue: 2 year: 2020 ident: 2029_CR8 publication-title: SIAM J. Discrete Math. doi: 10.1137/19M1299359 – volume: 30 start-page: 1 issue: 1 year: 2000 ident: 2029_CR21 publication-title: Integr. VLSI J. doi: 10.1016/S0167-9260(00)00008-0 – volume: 88 start-page: 507 issue: 3 year: 2000 ident: 2029_CR24 publication-title: Math. Program. doi: 10.1007/PL00011383 – ident: 2029_CR16 doi: 10.1109/FOCS.2018.00020 – volume: 90 start-page: 63 issue: 10 year: 2007 ident: 2029_CR22 publication-title: Electron. Commun. Jpn. doi: 10.1002/ecjc.20341 – volume: 102 start-page: 167 issue: 1 year: 2005 ident: 2029_CR35 publication-title: Math. Program. doi: 10.1007/s10107-004-0510-2 – volume: 10 start-page: 371 issue: 4 year: 2013 ident: 2029_CR11 publication-title: Discrete Optim. doi: 10.1016/j.disopt.2013.10.002 – volume: 3 start-page: 183 issue: 2/3 year: 1999 ident: 2029_CR12 publication-title: J. Comb. Optim. doi: 10.1023/A:1009833626004 – year: 2021 ident: 2029_CR3 publication-title: Math. Program. doi: 10.1007/s10107-021-01732-0 – volume: 47 start-page: 2118 issue: 6 year: 2018 ident: 2029_CR9 publication-title: SIAM J. Comput. doi: 10.1137/18M1163865 – ident: 2029_CR2 doi: 10.1137/1.9781611977073 – start-page: 208 volume-title: Algorithms and Computation year: 2008 ident: 2029_CR31 doi: 10.1007/978-3-540-92182-0_21 – volume: 56 start-page: 297 issue: 3 year: 2010 ident: 2029_CR33 publication-title: Algorithmica doi: 10.1007/s00453-009-9284-5 – ident: 2029_CR29 doi: 10.1145/1374376.1374402 – volume: 36 start-page: 1329 issue: 5 year: 2006 ident: 2029_CR18 publication-title: SIAM J. Comput. doi: 10.1137/050631616 – volume: 82 start-page: 3 issue: 1 year: 1998 ident: 2029_CR28 publication-title: Math. Program. doi: 10.1007/BF01585863 – year: 2022 ident: 2029_CR5 publication-title: Math. Oper. Res. doi: 10.1287/moor.2021.1250 – ident: 2029_CR19
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Snippet	In this paper, we study the minimum k -partition problem of submodular functions, i.e., given a finite set V and a submodular function f : 2 V → R , computing... In this paper, we study the minimum k-partition problem of submodular functions, i.e., given a finite set V and a submodular function [Formula omitted],...
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SubjectTerms	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical
Title	A polynomial time algorithm for finding a minimum 4-partition of a submodular function
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