Greedy algorithm for maximization of semi-monotone non-submodular functions with applications

The problem of maximizing submodular set functions has received increasing attention in recent years, and significant improvements have been made, particularly in relation to objective functions that satisfy monotonic submodularity. However, in practice, the objective function may not be monotonically submodular. While greedy algorithms have strong theoretical guarantees for maximizing submodular functions, their performance is barely guaranteed for non-submodular functions. Therefore, in this paper, we investigate the problem of maximizing non-monotone non-submodular functions under knapsack constraints based on the problem of infectious diseases and provides a more sophisticated analysis through the idea of segmentation. Since our definition characterizes the function more elaborately, a better bound, i.e., a tighter approximation guarantee, is achieved. Finally, we generalize the relevant results for the more general problems.