Algorithms for maximizing monotone submodular function minus modular function under noise

Submodular function has the property of diminishing marginal gain, and thus it has a wide range of applications in combinatorial optimization and in emerging disciplines such as machine learning and artificial intelligence. For any set S, most of previous works usually do not consider how to compute f(S) , but assume that there exists an oracle that will output f(S) directly. In reality, however, the process of computing the exact f is often inevitably inaccurate or costly. At this point, we adopt the easily available noise version F of f. In this paper, we investigate the problems of maximizing a non-negative monotone normalized submodular function minus a non-negative modular function under the $$\varepsilon $$ ε -multiplicative noise in three situations, i.e., the cardinality constraint, the matroid constraint and the online unconstraint. For the above problems, we design three deterministic bicriteria approximation algorithms using greedy and threshold ideas and furthermore obtain good approximation guarantees.