A constrained two-stage submodular maximization

Abstract In this paper, we investigate the two-stage submodular maximization problem, where there is a collection F = { f 1 , . . . , f m } of m submodular functions which are defined on the same element ground set Ω. The goal is to select a subset S ⊆ Ω of size at most l such that 1 m ∑ f ∈ F max T ⊆ S , T ∈ I ⁡ f ( T ) is maximized, where I denotes a specific-defined independence system. We consider the two-stage submodular maximization with a P-matroid constraint and present a ( 1 / ( P + 1 ) ) ( 1 − 1 / e ( P + 1 ) ) -approximation algorithm. Furthermore, we extend the algorithm to the two-stage submodular maximization with a more generalized P-exchange system constraint and show the approximation ratio can be maintained with slightly modifications of the algorithm.