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...
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| Published in: | Discrete Applied Mathematics Vol. 377; pp. 113 - 133 |
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31.12.2025
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| 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. |
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| 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 |
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| 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 |
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| Keywords | Greedy algorithm Submodular function Iterated submodular+modular procedure Supermodular function Sandwich method |
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| Title | Submodular + Supermodular function maximization with knapsack constraint |
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