Non-monotone submodular function maximization under k-system constraint

The problems of maximizing constrained monotone submodular functions have many practical applications, most recently in the context of combinatorial optimization, operations research, economics and especially machine learning, with constant approximation algorithms known under a variety of constrain...

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Published in:	Journal of combinatorial optimization Vol. 41; no. 1; pp. 128 - 142
Main Authors:	Shi, Majun, Yang, Zishen, Kim, Donghyun, Wang, Wei
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.01.2021 Springer Nature B.V
Subjects:	Algorithms Approximation Combinatorial analysis Combinatorics Computer science Constraints Convex and Discrete Geometry Economics Foundations Greedy algorithms Machine learning Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Maximization Operations research Operations Research/Decision Theory Optimization Run time (computers) Theory of Computation Modified-Greedy algorithm system Submodular maximization NMSFMk algorithm
ISSN:	1382-6905, 1573-2886
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Abstract	The problems of maximizing constrained monotone submodular functions have many practical applications, most recently in the context of combinatorial optimization, operations research, economics and especially machine learning, with constant approximation algorithms known under a variety of constraints. Unfortunately, non-monotone submodular functions maximization is less well studied; the first approximation algorithm for the non-monotone case was studied by Feige et al. (Proceedings of the 48th IEEE symposium on foundations of computer science (FOCS’07), 2007) about unconstrained non-monotone submodular maximization in 2007. In this paper, we extend the work of Lee et al. (Proceedings of the 41st ACM-SIAM symposium on theory of computing (STOC’09), pp 323–332, 2009) for maximizing a non-monotone submodular function under k -matroid constraint to k -system constraint. We first propose a Modified-Greedy algorithm that works no worse than that of Gupta et al. (Proceedings of the 6th international workshop on internet and network economics (WINE’10), vol 6484, pp 246–257, 2010). Based on this, then we provide the NMSFMk algorithm for maximizing a non-monotone submodular function subject to k -system constraint (which generalizes the k -matroid constraint), using Modified-Greedy algorithm combined with USFM algorithm (USFM algorithm is the random linear time 1/2-approximation algorithm proposed by Buchbinder et al. (Proceedings of the 53rd IEEE symposium on foundations of computer science (FOCS’12), pp 649–658, 2012) for unconstrained non-monotone submodular function maximization problem.) iteratively. Finally, we show that NMSFMk algorithm achieves a 1 2 k + 3 + 1 / k -approximation ratio with running time of O ( nmk ) (where m is the size of largest set returned by the NMSFMk algorithm), which beats the existing algorithms in many aspects.
AbstractList	The problems of maximizing constrained monotone submodular functions have many practical applications, most recently in the context of combinatorial optimization, operations research, economics and especially machine learning, with constant approximation algorithms known under a variety of constraints. Unfortunately, non-monotone submodular functions maximization is less well studied; the first approximation algorithm for the non-monotone case was studied by Feige et al. (Proceedings of the 48th IEEE symposium on foundations of computer science (FOCS’07), 2007) about unconstrained non-monotone submodular maximization in 2007. In this paper, we extend the work of Lee et al. (Proceedings of the 41st ACM-SIAM symposium on theory of computing (STOC’09), pp 323–332, 2009) for maximizing a non-monotone submodular function under k -matroid constraint to k -system constraint. We first propose a Modified-Greedy algorithm that works no worse than that of Gupta et al. (Proceedings of the 6th international workshop on internet and network economics (WINE’10), vol 6484, pp 246–257, 2010). Based on this, then we provide the NMSFMk algorithm for maximizing a non-monotone submodular function subject to k -system constraint (which generalizes the k -matroid constraint), using Modified-Greedy algorithm combined with USFM algorithm (USFM algorithm is the random linear time 1/2-approximation algorithm proposed by Buchbinder et al. (Proceedings of the 53rd IEEE symposium on foundations of computer science (FOCS’12), pp 649–658, 2012) for unconstrained non-monotone submodular function maximization problem.) iteratively. Finally, we show that NMSFMk algorithm achieves a 1 2 k + 3 + 1 / k -approximation ratio with running time of O ( nmk ) (where m is the size of largest set returned by the NMSFMk algorithm), which beats the existing algorithms in many aspects. The problems of maximizing constrained monotone submodular functions have many practical applications, most recently in the context of combinatorial optimization, operations research, economics and especially machine learning, with constant approximation algorithms known under a variety of constraints. Unfortunately, non-monotone submodular functions maximization is less well studied; the first approximation algorithm for the non-monotone case was studied by Feige et al. (Proceedings of the 48th IEEE symposium on foundations of computer science (FOCS’07), 2007) about unconstrained non-monotone submodular maximization in 2007. In this paper, we extend the work of Lee et al. (Proceedings of the 41st ACM-SIAM symposium on theory of computing (STOC’09), pp 323–332, 2009) for maximizing a non-monotone submodular function under k-matroid constraint to k-system constraint. We first propose a Modified-Greedy algorithm that works no worse than that of Gupta et al. (Proceedings of the 6th international workshop on internet and network economics (WINE’10), vol 6484, pp 246–257, 2010). Based on this, then we provide the NMSFMk algorithm for maximizing a non-monotone submodular function subject to k-system constraint (which generalizes the k-matroid constraint), using Modified-Greedy algorithm combined with USFM algorithm (USFM algorithm is the random linear time 1/2-approximation algorithm proposed by Buchbinder et al. (Proceedings of the 53rd IEEE symposium on foundations of computer science (FOCS’12), pp 649–658, 2012) for unconstrained non-monotone submodular function maximization problem.) iteratively. Finally, we show that NMSFMk algorithm achieves a 12k+3+1/k-approximation ratio with running time of O(nmk) (where m is the size of largest set returned by the NMSFMk algorithm), which beats the existing algorithms in many aspects.
Author	Kim, Donghyun Wang, Wei Yang, Zishen Shi, Majun
Author_xml	– sequence: 1 givenname: Majun surname: Shi fullname: Shi, Majun organization: School of Mathematics and Statistics, Xi’an Jiaotong University – sequence: 2 givenname: Zishen surname: Yang fullname: Yang, Zishen organization: School of Mathematics and Statistics, Xi’an Jiaotong University – sequence: 3 givenname: Donghyun surname: Kim fullname: Kim, Donghyun organization: Department of Computer Science, Georgia State University – sequence: 4 givenname: Wei surname: Wang fullname: Wang, Wei email: wang_weiw@xjtu.edu.cn organization: School of Mathematics and Statistics, Xi’an Jiaotong University
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References_xml	– volume: 9 start-page: 235 issue: 3 year: 2008 end-page: 284 ident: CR16 article-title: Near-optimal sensor placements in gaussian processes: theory, efficient algorithms and empirical studies publication-title: J Mach Learn Res – ident: CR19 – volume: 23 start-page: 2053 year: 2010 end-page: 2078 ident: CR18 article-title: Maximizing non-monotone submodular functions under matroid or knapsack constraints publication-title: SIAM J Discrete Math doi: 10.1137/090750020 – volume: 14 start-page: 265 issue: 1 year: 1978 end-page: 294 ident: CR21 article-title: An analysis of approximations for maximizing submodular set functions-I publication-title: Math Program doi: 10.1007/BF01588971 – volume: 40 start-page: 1740 issue: 6 year: 2011 end-page: 1766 ident: CR4 article-title: Maximizing a monotone submodular function subject to a matroid constraint publication-title: SIAM J Comput doi: 10.1137/080733991 – ident: CR3 – ident: CR14 – ident: CR2 – ident: CR17 – ident: CR13 – volume: 70 start-page: 39 issue: 1 year: 1999 end-page: 45 ident: CR15 article-title: The budgeted maximum coverage problem publication-title: Inf Process Lett doi: 10.1016/S0020-0190(99)00031-9 – ident: CR9 – volume: 14 start-page: 1 year: 2018 end-page: 20 ident: CR1 article-title: Deterministic algorithms for submodular maximization problems publication-title: ACM Trans Algorithms doi: 10.1145/3184990 – volume: 8 start-page: 73 year: 1978 end-page: 87 ident: CR11 article-title: An analysis of approximations for maximizing submodular set functions-II publication-title: Math Prog Study doi: 10.1007/BFb0121195 – ident: CR6 – volume: 43 start-page: 514 issue: 2 year: 2013 end-page: 542 ident: CR10 article-title: Monotone submodular maximization over a matroid via non-oblivious local search publication-title: SIAM J Comput doi: 10.1137/130920277 – ident: CR5 – ident: CR7 – ident: CR8 – volume: 32 start-page: 41 issue: 1 year: 2004 end-page: 43 ident: CR22 article-title: A note on maximizing a submodular set function subject to a knapsack constraint publication-title: Oper Res Lett doi: 10.1016/S0167-6377(03)00062-2 – year: 2005 ident: CR12 publication-title: Submodular functions and optimization – ident: CR24 – ident: CR23 – ident: CR20 – volume: 14 start-page: 1 year: 2018 ident: 672_CR1 publication-title: ACM Trans Algorithms doi: 10.1145/3184990 – ident: 672_CR3 doi: 10.1137/1.9781611973402.106 – volume-title: Submodular functions and optimization year: 2005 ident: 672_CR12 – ident: 672_CR23 doi: 10.1145/1374376.1374389 – ident: 672_CR24 doi: 10.1109/FOCS.2009.24 – volume: 23 start-page: 2053 year: 2010 ident: 672_CR18 publication-title: SIAM J Discrete Math doi: 10.1137/090750020 – ident: 672_CR7 doi: 10.1109/FOCS.2007.4389516 – ident: 672_CR2 doi: 10.1109/FOCS.2012.73 – ident: 672_CR5 doi: 10.1007/978-3-540-72792-7_15 – ident: 672_CR8 – ident: 672_CR6 – ident: 672_CR19 doi: 10.1145/1536414.1536459 – ident: 672_CR20 – volume: 43 start-page: 514 issue: 2 year: 2013 ident: 672_CR10 publication-title: SIAM J Comput doi: 10.1137/130920277 – volume: 70 start-page: 39 issue: 1 year: 1999 ident: 672_CR15 publication-title: Inf Process Lett doi: 10.1016/S0020-0190(99)00031-9 – ident: 672_CR17 doi: 10.1137/1.9781611973068.60 – volume: 14 start-page: 265 issue: 1 year: 1978 ident: 672_CR21 publication-title: Math Program doi: 10.1007/BF01588971 – volume: 32 start-page: 41 issue: 1 year: 2004 ident: 672_CR22 publication-title: Oper Res Lett doi: 10.1016/S0167-6377(03)00062-2 – ident: 672_CR9 doi: 10.1109/FOCS.2011.46 – volume: 8 start-page: 73 year: 1978 ident: 672_CR11 publication-title: Math Prog Study doi: 10.1007/BFb0121195 – ident: 672_CR13 – volume: 40 start-page: 1740 issue: 6 year: 2011 ident: 672_CR4 publication-title: SIAM J Comput doi: 10.1137/080733991 – volume: 9 start-page: 235 issue: 3 year: 2008 ident: 672_CR16 publication-title: J Mach Learn Res – ident: 672_CR14 doi: 10.1007/978-3-642-17572-5_20
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SubjectTerms	Algorithms Approximation Combinatorial analysis Combinatorics Computer science Constraints Convex and Discrete Geometry Economics Foundations Greedy algorithms Machine learning Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Maximization Operations research Operations Research/Decision Theory Optimization Run time (computers) Theory of Computation
Title	Non-monotone submodular function maximization under k-system constraint
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