Submodular + Supermodular function maximization with knapsack constraint - Shi, Majun; Yang, Zishen; Wang, Wei (Discrete Applied Mathematics, 2025)
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.