Fast deterministic algorithms for non-submodular maximization with strong performance guarantees

We study the non-submodular maximization problem, in which the objective function is characterized by parameters, subject to a cardinality or \(p\)-system constraint. By adapting the Threshold-Greedy algorithm for the submodular maximization, we present two deterministic algorithms for approximately solving the non-submodular maximization problem. Our analysis shows that the algorithms we propose requires much less function evaluations than existing algorithms, while providing comparable approximation guarantees. Moreover, numerical experiment results are presented to validate the theoretical analysis. Our results not only fill a gap in the (non-)submodular maximization, but also generalize and improve several existing results on closely related optimization problems.