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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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 377; pp. 113 - 133
Main Authors: Shi, Majun, Yang, Zishen, Wang, Wei
Format: Journal Article
Language:English
Published: Elsevier B.V 31.12.2025
Subjects:
Greedy algorithm
Iterated submodular+modular procedure
Sandwich method
Submodular function
Supermodular function
Greedy algorithm
Submodular function
Iterated submodular+modular procedure
Supermodular function
Sandwich method
ISSN:0166-218X
Online Access:Get full text
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Summary: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.
ISSN:0166-218X
DOI:10.1016/j.dam.2025.06.062