Maximizing a monotone non-submodular function under a knapsack constraint

• Published in: Journal of combinatorial optimization Vol. 43; no. 5; pp. 1125 - 1148
• Main Authors: Zhang, Zhenning, Liu, Bin, Wang, Yishui, Xu, Dachuan, Zhang, Dongmei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2022; Springer Nature B.V
• Subjects: Combinatorial analysis; Combinatorics; Convex and Discrete Geometry; Design of experiments; Design optimization; Greedy algorithms; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Maximization; Operations Research/Decision Theory; Optimization; Social networks; Theory of Computation; Submodularity ratio; Diminishing-return ratio; Curvature; Non-submodular; Knapsack constraint
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-020-00620-1

Submodular optimization has been well studied in combinatorial optimization. However, there are few works considering about non-submodular optimization problems which also have many applications, such as experimental design, some optimization problems in social networks, etc. In this paper, we consider the maximization of non-submodular function under a knapsack constraint, and explore the performance of the greedy algorithm, which is characterized by the submodularity ratio β and curvature α . In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of ( 1 - e - α β ) / α for the above problem. To our knowledge, it is the first tight constant factor for this problem. We further utilize illustrative examples to demonstrate the performance of our algorithm.