An accelerated deterministic algorithm for maximizing monotone submodular minus modular function with cardinality constraint

Submodular optimization not only covers some classical combinatorial optimization problems, but also has a wide range of applications in fields such as machine learning and artificial intelligence. For submodular maximization problems with constraints, some work has been done including the design of approximation algorithms, the measurement of approximation algorithms in terms of quality and efficiency, etc. In this paper, we consider the problem of maximizing a non-negative monotone submodular function minus a non-negative modular function with the cardinality constraint. This model has been applied to many scenarios, such as team formation problem, influence maximization problem, recommender systems problem, etc. We propose a threshold algorithm that achieve a $(1/2-O(\epsilon ),2)$-bicriteria approximation ratio and query complexity $O(n\log n)$. Our algorithm makes a small sacrifice in the approximation ratio but improves the best query complexity result of existing deterministic algorithms from $O\left(n^2\right)$ to $O(n\log n)$ in the worst case.