Maximizing a non-decreasing non-submodular function subject to various types of constraints

• Published in: Journal of global optimization Vol. 83; no. 4; pp. 727 - 751
• Main Authors: Lu, Cheng, Yang, Wenguo, Yang, Ruiqi, Gao, Suixiang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computer Science; Design; Machine learning; Mathematical analysis; Mathematics; Mathematics and Statistics; Maximization; Operations Research/Decision Theory; Optimization; Real Functions; Search algorithms; Local search; Supermodular function; Non-submodular maximization; Weakly DR-submodular function; Curvature; Matroid
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-021-01123-x

In this paper, we firstly study the problem of maximizing a γ -weakly DR-submodular function under a general matroid constraint. We present a local search algorithm, which is guided by a tailored potential function, for solving this problem. We prove that our algorithm produces a ( 1 - e - γ - ϵ )-approximate solution. To the best of our knowledge, it’s the first algorithm achieving the tight approximation guarantee for such maximization problem. In addition, we study the maximization of the sum of submodular and supermodular functions. We show that this problem can be reduced to the maximization of submodular and linear sums. Based on this reduction, we derive new and improved approximation bounds for the problem under various types of constraints.