Submodular + Supermodular function maximization with knapsack constraint
We investigate a class of non-submodular function optimization problems, specifically maximizing the sum of a normalized monotone submodular function f and a normalized monotone supermodular function g under a knapsack constraint. By utilizing the total curvature κf of f and the supermodular curvatu...
Gespeichert in:
| Veröffentlicht in: | Discrete Applied Mathematics Jg. 377; S. 113 - 133 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Elsevier B.V
31.12.2025
|
| Schlagworte: | |
| ISSN: | 0166-218X |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | We investigate a class of non-submodular function optimization problems, specifically maximizing the sum of a normalized monotone submodular function f and a normalized monotone supermodular function g under a knapsack constraint. By utilizing the total curvature κf of f and the supermodular curvature κg of g, we demonstrate that this problem can achieve a near-optimal solution through three approaches: a greedy algorithm, an iterated submodular+modular procedure and a sandwich method. In particular, we prove that both the greedy algorithm and the iterated submodular+modular procedure provide an approximation guarantee of 1κf(1−e−(1−κg)κf), while the sandwich method achieves a (1−κg)(1−κfe)-approximation ratio. All proposed algorithms run in polynomial time, and parameters such as κf and κg can be computed efficiently in linear time. Additionally, all three algorithms yield a (1−κg)-approximation performance for knapsack-constrained monotone supermodular function maximization. Finally, we empirically test our first two algorithms on a constructed application. Although both algorithms have the same theoretical guarantee, their practical behavior differs significantly, leading to distinct solutions. |
|---|---|
| AbstractList | We investigate a class of non-submodular function optimization problems, specifically maximizing the sum of a normalized monotone submodular function f and a normalized monotone supermodular function g under a knapsack constraint. By utilizing the total curvature κf of f and the supermodular curvature κg of g, we demonstrate that this problem can achieve a near-optimal solution through three approaches: a greedy algorithm, an iterated submodular+modular procedure and a sandwich method. In particular, we prove that both the greedy algorithm and the iterated submodular+modular procedure provide an approximation guarantee of 1κf(1−e−(1−κg)κf), while the sandwich method achieves a (1−κg)(1−κfe)-approximation ratio. All proposed algorithms run in polynomial time, and parameters such as κf and κg can be computed efficiently in linear time. Additionally, all three algorithms yield a (1−κg)-approximation performance for knapsack-constrained monotone supermodular function maximization. Finally, we empirically test our first two algorithms on a constructed application. Although both algorithms have the same theoretical guarantee, their practical behavior differs significantly, leading to distinct solutions. |
| Author | Wang, Wei Yang, Zishen Shi, Majun |
| Author_xml | – sequence: 1 givenname: Majun orcidid: 0000-0002-7990-9959 surname: Shi fullname: Shi, Majun organization: School of Mathematics, Xi’an University of Finance and Economics, Xi’an, 710100, China – sequence: 2 givenname: Zishen surname: Yang fullname: Yang, Zishen organization: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China – sequence: 3 givenname: Wei surname: Wang fullname: Wang, Wei email: wang_weiw@163.com organization: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China |
| BookMark | eNp9j09LAzEQxXOoYFv9AN72LrtO0nZ3gycpaoWChyp4C9NJgmm72ZLs-u_Tm1q9Cg-G9-AN7zdiA996w9gFh4IDL682hcamECBmBZRJYsCGKS9zweuXUzaKcQMAPLkhW6z6ddPqfochu8xW_d6EP2t7T51rfdbgh2vcF_6Yd9e9ZluP-4i0zaj1sQvofHfGTizuojn_vWP2fHf7NF_ky8f7h_nNMqe0oMsNTGaCpABLUElEU2JtpEY-parUhIZL4LqyUysIrFzXEiqYgBC6trLCejJm_PiXQhtjMFbtg2swfCoO6oCvNirhqwO-gjJJpM71sWPSsDdngorkjCejXTDUKd26f9rfqwhoNg |
| Cites_doi | 10.1016/S0020-0190(99)00031-9 10.1007/BF01588971 10.1016/j.disc.2024.114287 10.1007/978-3-030-27195-4_12 10.1016/0166-218X(84)90003-9 10.1007/978-3-540-72792-7_15 10.1016/j.dam.2015.01.026 10.1007/BFb0121195 10.1007/s10898-021-01063-6 10.1016/j.tcs.2020.05.018 10.1007/s10957-022-02145-5 10.1137/130920277 10.1137/16M1107644 10.1016/S0167-6377(03)00062-2 10.1145/1374376.1374389 10.1007/978-3-030-26176-4_54 10.1007/s40305-018-0233-3 10.1137/080733991 10.1137/1.9781611973730.76 10.1007/s10898-021-01123-x 10.1007/s40305-022-00444-2 |
| ContentType | Journal Article |
| Copyright | 2025 Elsevier B.V. |
| Copyright_xml | – notice: 2025 Elsevier B.V. |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.dam.2025.06.062 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EndPage | 133 |
| ExternalDocumentID | 10_1016_j_dam_2025_06_062 S0166218X25003786 |
| GrantInformation_xml | – fundername: National Natural Science Foundation of China grantid: 11971376 funderid: http://dx.doi.org/10.13039/501100001809 – fundername: Young Talent Fund of Association for Science and Technology in Shaanxi grantid: 20230506 – fundername: Scientific Research Program Funded by Education Department of Shaanxi Provincial Government grantid: 23JK0449 – fundername: Natural Science Basic Research Program of Shaanxi grantid: 2023-JC-QN-0086 |
| GroupedDBID | -~X ADEZE AFTJW ALMA_UNASSIGNED_HOLDINGS FDB OAUVE AAYXX AI. CITATION FA8 VH1 WUQ |
| ID | FETCH-LOGICAL-c166t-e0352c920fc079aae6a8e9da14c76dcae1901d7f4f2c0f9b890703022d8f97a83 |
| ISSN | 0166-218X |
| IngestDate | Wed Nov 05 20:55:42 EST 2025 Sat Oct 11 16:52:43 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Greedy algorithm Submodular function Iterated submodular+modular procedure Supermodular function Sandwich method |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c166t-e0352c920fc079aae6a8e9da14c76dcae1901d7f4f2c0f9b890703022d8f97a83 |
| ORCID | 0000-0002-7990-9959 |
| PageCount | 21 |
| ParticipantIDs | crossref_primary_10_1016_j_dam_2025_06_062 elsevier_sciencedirect_doi_10_1016_j_dam_2025_06_062 |
| PublicationCentury | 2000 |
| PublicationDate | 2025-12-31 |
| PublicationDateYYYYMMDD | 2025-12-31 |
| PublicationDate_xml | – month: 12 year: 2025 text: 2025-12-31 day: 31 |
| PublicationDecade | 2020 |
| PublicationTitle | Discrete Applied Mathematics |
| PublicationYear | 2025 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | Filmus, Ward (b9) 2013; 43 Nemhauser, Wolsey, Fisher (b17) 1978; 14 Liu, Guo, Du, Xu, Zhang (b14) 2022; 82 Z.N. Zhang, B. Liu, Y.S. Wang, D.C. Xu, D.M. Zhang, A greedy algorithm for maximization of non-submodular functions subject to a knapsack constraint, in: Proceedings of the 26th International Computing and Combinatorics Conference, 2019, pp. 651–662. Lu, Yang, Yang, Gao (b15) 2022; 83 M. Narasimhan, J. Bilmes, A submodular-supermodular procedure with applications to discriminative structure learning, in: Proceedings of the 21st Conference on Uncertainty in Artificial Intelligence, 2005, pp. 404–412. Khuller, Moss, Naor (b13) 1999; 70 M. Sviridenko, J. Vondrák, J. Ward, Optimal approximation for submodular and supermodular optimization with bounded curvature, in: Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms, 2015, pp. 1134–1148. Fisher, Nemhauser, Wolsey (b10) 1978; 8 Sviridenko (b22) 2004; 32 Shi, Yang, Wang (b20) 2023; 196 R. Iyer, J. Bilmes, Algorithms for approximate minimization of the difference between submodular functions, with applications, in: Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence, 2012, pp. 407–417. C.G. Gao, S.Y. Gu, R.Q. Yang, J.G. Yu, W.L. Wu, D.C. Xu, Interaction-aware influence maximization and iterated sandwich method, in: Proceedings of the 13th International Conference on Algorithmic Aspects in Information and Management, 2019, pp. 129–141. Nong, Sun, Gong, Fang, Du, Shao (b18) 2021; 853 A.A. Bian, J.M. Buhmann, A. Krause, S. Tschiatschek, Guarantees for greedy maximization of non-submodular functions with applications, in: Proceedings of the 34th International Conference on Machine Learning, 2017, pp. 498–507. Byrnes (b3) 2015; 186 Conforti, Cornuéjols (b7) 1984; 7 Vondrák (b25) 2009; B23 G. Calinescu, C. Chekuri, M. Pál, J. Vondrák, Maximizing a submodular set function subject to a matroid constraint (extended abstract), in: Proceedings of the 12th Conference on Integer Programming and Combinatorial Optimization, vol. 4513, 2007, pp. 182–196. Calinescu, Chekuri, Pál, Vondrák (b5) 2011; 40 Yoshida (b27) 2019; 33 W. Bai, J.A. Bilmes, Greed is still good: maximizing monotone submodular+supermodular functions, in: Proceedings of the 35th International Conference on Machine Learning, 2018, pp. 304–313. L. Chen, M. Feldman, A. Karbasi, Weakly submodular maximization beyond cardinality constraints: does randomization help greedy?, in: Proceedings of the 35th International Conference on Machine Learning, 2018, pp. 804–813. J. Vondrák, Optimal approximation for the submodular welfare problem in the value oracle model, in: Proceedings of the 40th ACM Symposium on Theory of Computing, 2008, pp. 67–74. Shi, Zhu, Liu, Li (b21) 2025; 348 Shi, Wang (b19) 2024; 12 A. Das, D. Kempe, Submodular meets spectral: greedy algorithms for subset selection, sparse approximation, and dictionary selection, in: Proceedings of the 28th International Conference on Machine Learning, 2011, pp. 1057–1064. Wu, Zhang, Du (b26) 2019; 7 Fisher (10.1016/j.dam.2025.06.062_b10) 1978; 8 10.1016/j.dam.2025.06.062_b16 Filmus (10.1016/j.dam.2025.06.062_b9) 2013; 43 10.1016/j.dam.2025.06.062_b11 Vondrák (10.1016/j.dam.2025.06.062_b25) 2009; B23 10.1016/j.dam.2025.06.062_b12 Shi (10.1016/j.dam.2025.06.062_b21) 2025; 348 Sviridenko (10.1016/j.dam.2025.06.062_b22) 2004; 32 Lu (10.1016/j.dam.2025.06.062_b15) 2022; 83 Shi (10.1016/j.dam.2025.06.062_b20) 2023; 196 Nemhauser (10.1016/j.dam.2025.06.062_b17) 1978; 14 10.1016/j.dam.2025.06.062_b2 Nong (10.1016/j.dam.2025.06.062_b18) 2021; 853 10.1016/j.dam.2025.06.062_b4 Khuller (10.1016/j.dam.2025.06.062_b13) 1999; 70 Conforti (10.1016/j.dam.2025.06.062_b7) 1984; 7 10.1016/j.dam.2025.06.062_b1 10.1016/j.dam.2025.06.062_b28 Calinescu (10.1016/j.dam.2025.06.062_b5) 2011; 40 10.1016/j.dam.2025.06.062_b6 10.1016/j.dam.2025.06.062_b24 10.1016/j.dam.2025.06.062_b8 Byrnes (10.1016/j.dam.2025.06.062_b3) 2015; 186 Yoshida (10.1016/j.dam.2025.06.062_b27) 2019; 33 10.1016/j.dam.2025.06.062_b23 Liu (10.1016/j.dam.2025.06.062_b14) 2022; 82 Wu (10.1016/j.dam.2025.06.062_b26) 2019; 7 Shi (10.1016/j.dam.2025.06.062_b19) 2024; 12 |
| References_xml | – reference: W. Bai, J.A. Bilmes, Greed is still good: maximizing monotone submodular+supermodular functions, in: Proceedings of the 35th International Conference on Machine Learning, 2018, pp. 304–313. – reference: M. Sviridenko, J. Vondrák, J. Ward, Optimal approximation for submodular and supermodular optimization with bounded curvature, in: Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms, 2015, pp. 1134–1148. – reference: J. Vondrák, Optimal approximation for the submodular welfare problem in the value oracle model, in: Proceedings of the 40th ACM Symposium on Theory of Computing, 2008, pp. 67–74. – volume: 853 start-page: 16 year: 2021 end-page: 24 ident: b18 article-title: Maximize a monotone function with a generic submodularity ratio publication-title: Theory Comput. Sci. – volume: B23 start-page: 253 year: 2009 end-page: 266 ident: b25 article-title: Submodularity and curvature: the optimal algorithm publication-title: RIMS Kokyuroku Bessatsu – volume: 196 start-page: 516 year: 2023 end-page: 543 ident: b20 article-title: Greedy guarantees for non-submodular function maximization under independent system constraint with applications publication-title: J. Optim. Theory Appl. – volume: 40 start-page: 1740 year: 2011 end-page: 1766 ident: b5 article-title: Maximizing a monotone submodular function subject to a matroid constraint publication-title: SIAM J. Comput. – volume: 70 start-page: 39 year: 1999 end-page: 45 ident: b13 article-title: The budgeted maximum coverage problem publication-title: Inform. Process. Lett. – volume: 7 start-page: 183 year: 2019 end-page: 193 ident: b26 article-title: Set function optimization publication-title: J. Oper. Res. Soc. China – reference: G. Calinescu, C. Chekuri, M. Pál, J. Vondrák, Maximizing a submodular set function subject to a matroid constraint (extended abstract), in: Proceedings of the 12th Conference on Integer Programming and Combinatorial Optimization, vol. 4513, 2007, pp. 182–196. – reference: A.A. Bian, J.M. Buhmann, A. Krause, S. Tschiatschek, Guarantees for greedy maximization of non-submodular functions with applications, in: Proceedings of the 34th International Conference on Machine Learning, 2017, pp. 498–507. – volume: 186 start-page: 275 year: 2015 end-page: 282 ident: b3 article-title: A tight analysis of the submodular-supermodular procedure publication-title: Discrete Appl. Math. – volume: 12 start-page: 627 year: 2024 end-page: 648 ident: b19 article-title: Greedy is good: constrained non-submodular function maximization via weak submodularity publication-title: J. Oper. Res. Soc. China – volume: 32 start-page: 41 year: 2004 end-page: 43 ident: b22 article-title: A note on maximizing a submodular set function subject to a knapsack constraint publication-title: Oper. Res. Lett. – reference: L. Chen, M. Feldman, A. Karbasi, Weakly submodular maximization beyond cardinality constraints: does randomization help greedy?, in: Proceedings of the 35th International Conference on Machine Learning, 2018, pp. 804–813. – volume: 14 start-page: 265 year: 1978 end-page: 294 ident: b17 article-title: An analysis of approximations for maximizing submodular set functions-I publication-title: Math. Program. – reference: A. Das, D. Kempe, Submodular meets spectral: greedy algorithms for subset selection, sparse approximation, and dictionary selection, in: Proceedings of the 28th International Conference on Machine Learning, 2011, pp. 1057–1064. – reference: C.G. Gao, S.Y. Gu, R.Q. Yang, J.G. Yu, W.L. Wu, D.C. Xu, Interaction-aware influence maximization and iterated sandwich method, in: Proceedings of the 13th International Conference on Algorithmic Aspects in Information and Management, 2019, pp. 129–141. – reference: M. Narasimhan, J. Bilmes, A submodular-supermodular procedure with applications to discriminative structure learning, in: Proceedings of the 21st Conference on Uncertainty in Artificial Intelligence, 2005, pp. 404–412. – volume: 7 start-page: 251 year: 1984 end-page: 274 ident: b7 article-title: Submodular functions, matroids, and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado-Edmonds theorem publication-title: Discrete Appl. Math. – volume: 43 start-page: 514 year: 2013 end-page: 542 ident: b9 article-title: Monotone submodular maximization over a matroid via non-oblivious local search publication-title: SIAM J. Comput. – volume: 348 year: 2025 ident: b21 article-title: Weak submodularity implies localizability: local search for constrained non-submodular function maximization publication-title: Discrete Math. – volume: 83 start-page: 727 year: 2022 end-page: 751 ident: b15 article-title: Maximizing a non-decreasing non-submodular function subject to various types of constraints publication-title: J. Global Optim. – reference: Z.N. Zhang, B. Liu, Y.S. Wang, D.C. Xu, D.M. Zhang, A greedy algorithm for maximization of non-submodular functions subject to a knapsack constraint, in: Proceedings of the 26th International Computing and Combinatorics Conference, 2019, pp. 651–662. – volume: 82 start-page: 179 year: 2022 end-page: 194 ident: b14 article-title: Maximization problems of balancing submodular relevance and supermodular diversity publication-title: J. Global Optim. – volume: 8 start-page: 73 year: 1978 end-page: 87 ident: b10 article-title: An analysis of approximations for maximizing submodular set functions-II publication-title: Math. Prog. Study – reference: R. Iyer, J. Bilmes, Algorithms for approximate minimization of the difference between submodular functions, with applications, in: Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence, 2012, pp. 407–417. – volume: 33 start-page: 1452 year: 2019 end-page: 1471 ident: b27 article-title: Maximizing a monotone submodular function with a bounded curvature under a knapsack constraint publication-title: SIAM J. Discrete Math. – volume: 70 start-page: 39 issue: 1 year: 1999 ident: 10.1016/j.dam.2025.06.062_b13 article-title: The budgeted maximum coverage problem publication-title: Inform. Process. Lett. doi: 10.1016/S0020-0190(99)00031-9 – volume: 14 start-page: 265 issue: 1 year: 1978 ident: 10.1016/j.dam.2025.06.062_b17 article-title: An analysis of approximations for maximizing submodular set functions-I publication-title: Math. Program. doi: 10.1007/BF01588971 – volume: 348 issue: 2 year: 2025 ident: 10.1016/j.dam.2025.06.062_b21 article-title: Weak submodularity implies localizability: local search for constrained non-submodular function maximization publication-title: Discrete Math. doi: 10.1016/j.disc.2024.114287 – ident: 10.1016/j.dam.2025.06.062_b1 – ident: 10.1016/j.dam.2025.06.062_b11 doi: 10.1007/978-3-030-27195-4_12 – volume: B23 start-page: 253 year: 2009 ident: 10.1016/j.dam.2025.06.062_b25 article-title: Submodularity and curvature: the optimal algorithm publication-title: RIMS Kokyuroku Bessatsu – volume: 7 start-page: 251 issue: 3 year: 1984 ident: 10.1016/j.dam.2025.06.062_b7 article-title: Submodular functions, matroids, and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado-Edmonds theorem publication-title: Discrete Appl. Math. doi: 10.1016/0166-218X(84)90003-9 – ident: 10.1016/j.dam.2025.06.062_b4 doi: 10.1007/978-3-540-72792-7_15 – volume: 186 start-page: 275 year: 2015 ident: 10.1016/j.dam.2025.06.062_b3 article-title: A tight analysis of the submodular-supermodular procedure publication-title: Discrete Appl. Math. doi: 10.1016/j.dam.2015.01.026 – volume: 8 start-page: 73 year: 1978 ident: 10.1016/j.dam.2025.06.062_b10 article-title: An analysis of approximations for maximizing submodular set functions-II publication-title: Math. Prog. Study doi: 10.1007/BFb0121195 – ident: 10.1016/j.dam.2025.06.062_b16 – volume: 82 start-page: 179 year: 2022 ident: 10.1016/j.dam.2025.06.062_b14 article-title: Maximization problems of balancing submodular relevance and supermodular diversity publication-title: J. Global Optim. doi: 10.1007/s10898-021-01063-6 – ident: 10.1016/j.dam.2025.06.062_b12 – volume: 853 start-page: 16 year: 2021 ident: 10.1016/j.dam.2025.06.062_b18 article-title: Maximize a monotone function with a generic submodularity ratio publication-title: Theory Comput. Sci. doi: 10.1016/j.tcs.2020.05.018 – volume: 196 start-page: 516 issue: 2 year: 2023 ident: 10.1016/j.dam.2025.06.062_b20 article-title: Greedy guarantees for non-submodular function maximization under independent system constraint with applications publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-022-02145-5 – ident: 10.1016/j.dam.2025.06.062_b2 – ident: 10.1016/j.dam.2025.06.062_b6 – volume: 43 start-page: 514 issue: 2 year: 2013 ident: 10.1016/j.dam.2025.06.062_b9 article-title: Monotone submodular maximization over a matroid via non-oblivious local search publication-title: SIAM J. Comput. doi: 10.1137/130920277 – volume: 33 start-page: 1452 year: 2019 ident: 10.1016/j.dam.2025.06.062_b27 article-title: Maximizing a monotone submodular function with a bounded curvature under a knapsack constraint publication-title: SIAM J. Discrete Math. doi: 10.1137/16M1107644 – ident: 10.1016/j.dam.2025.06.062_b8 – volume: 32 start-page: 41 issue: 1 year: 2004 ident: 10.1016/j.dam.2025.06.062_b22 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 – ident: 10.1016/j.dam.2025.06.062_b24 doi: 10.1145/1374376.1374389 – ident: 10.1016/j.dam.2025.06.062_b28 doi: 10.1007/978-3-030-26176-4_54 – volume: 7 start-page: 183 issue: 2 year: 2019 ident: 10.1016/j.dam.2025.06.062_b26 article-title: Set function optimization publication-title: J. Oper. Res. Soc. China doi: 10.1007/s40305-018-0233-3 – volume: 40 start-page: 1740 issue: 6 year: 2011 ident: 10.1016/j.dam.2025.06.062_b5 article-title: Maximizing a monotone submodular function subject to a matroid constraint publication-title: SIAM J. Comput. doi: 10.1137/080733991 – ident: 10.1016/j.dam.2025.06.062_b23 doi: 10.1137/1.9781611973730.76 – volume: 83 start-page: 727 year: 2022 ident: 10.1016/j.dam.2025.06.062_b15 article-title: Maximizing a non-decreasing non-submodular function subject to various types of constraints publication-title: J. Global Optim. doi: 10.1007/s10898-021-01123-x – volume: 12 start-page: 627 issue: 3 year: 2024 ident: 10.1016/j.dam.2025.06.062_b19 article-title: Greedy is good: constrained non-submodular function maximization via weak submodularity publication-title: J. Oper. Res. Soc. China doi: 10.1007/s40305-022-00444-2 |
| SSID | ssj0001218 ssj0000186 ssj0006644 |
| Score | 2.434592 |
| Snippet | We investigate a class of non-submodular function optimization problems, specifically maximizing the sum of a normalized monotone submodular function f and a... |
| SourceID | crossref elsevier |
| SourceType | Index Database Publisher |
| StartPage | 113 |
| SubjectTerms | Greedy algorithm Iterated submodular+modular procedure Sandwich method Submodular function Supermodular function |
| Title | Submodular + Supermodular function maximization with knapsack constraint |
| URI | https://dx.doi.org/10.1016/j.dam.2025.06.062 |
| Volume | 377 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 issn: 0166-218X databaseCode: AIEXJ dateStart: 20211209 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: false ssIdentifier: ssj0001218 providerName: Elsevier – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 issn: 0166-218X databaseCode: AIEXJ dateStart: 20220331 customDbUrl: isFulltext: true dateEnd: 99991231 titleUrlDefault: https://www.sciencedirect.com omitProxy: false ssIdentifier: ssj0001218 providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3PT9swFLa6skN3mAbbRDc2-bBdQJESN4ntY8c6AWJoEiAKl8h1bKktDRX9of75e47tNB0gsUm7RNWrEqd-X18-vzx_D6Ev8P8JTceBIGEyD-I00QEXHBYrilMN9CBVVjL_lJ6dsX6f_2o0TvxemOUtLQq2WvHpf3U12MDZZuvsX7i7uigY4DM4HY7gdjg-y_EQCiZ3eVld-pV8g8gwheDrDOYpVjp8IlbDiduCaXOx40JMZ0KOTR36rGwcsZG1_z6E-AIEu6KtPyu914qVn5cNguGb0aKC3LXLR9-Y2vvKeuWsV2pYTzuQxIsb-lzYg_0wNj2ZpgGQhn49vnZcnxYbISO79dQ9bCOrgvEgjtuUwgiGN2oBJCk1Vl3Y3pTHPjdjmiGBy4UdytIXaIvQhLMm2uoe9_onNTExo5TX8um39dsmYF2x04C3d-_ffpd1gH_cwuP8pcZJLt6g124xgbsWBNuooYod9KrmmbfoaA0HfIDrYMAeDLgOBmzAgD0Y8BoM79Dlj97F4VHgumcEEn7HPFBG6VZyEmoZUi6ESgVTPBdRLGmaS6EMFcypjjWRoeYDxsvoT0jONKeCdd6jZnFXqF2EozjSctCRsPiFBS4RPKGSUTLQsNZM4LpttO_nJJtakZTMVw-OMpjAzExgZiooU9JGsZ-1zLE8y94ycP7Tp334t9M-otYavXuoOb9fqE_opVzOh7P7zw4ivwEHz3Ni |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Submodular+%2B+Supermodular+function+maximization+with+knapsack+constraint&rft.jtitle=Discrete+Applied+Mathematics&rft.au=Shi%2C+Majun&rft.au=Yang%2C+Zishen&rft.au=Wang%2C+Wei&rft.date=2025-12-31&rft.pub=Elsevier+B.V&rft.issn=0166-218X&rft.volume=377&rft.spage=113&rft.epage=133&rft_id=info:doi/10.1016%2Fj.dam.2025.06.062&rft.externalDocID=S0166218X25003786 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0166-218X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0166-218X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0166-218X&client=summon |