Unconstrained Submodular Maximization with Modular Costs: Tight Approximation and Application to Profit Maximization

Given a set V , the problem of unconstrained submodular maximization with modular costs (USM-MC) asks for a subset S ⊆ V that maximizes f (S) − c (S), where f is a non-negative, monotone, and submodular function that gauges the utility of S , and c is a nonnegative and modular function that measures the cost of S . This problem finds applications in numerous practical scenarios, such as profit maximization in viral marketing on social media. This paper presents ROI-Greedy, a polynomial time algorithm for USM-MC that returns a solution S satisfying f (S) − c (S) ≥ f (S∗) − c (S∗) − ln f (S ∗) c (S∗) · c (S ∗), where S∗ is the optimal solution to USM-MC. To our knowledge, ROI-Greedy is the first algorithm that provides such a strong approximation guarantee. In addition, we show that this worst-case guarantee is tight, in the sense that no polynomial time algorithm can ensure f (S) − c (S) ≥ (1 + ε) · ( f (S∗) − c (S∗) − ln f (S ∗) c (S∗) · c (S ∗) ) , for any ε > 0. Further, we devise a non-trivial extension of ROI-Greedy to solve the profit maximization problem, where the precise value of f (S) for any set S is unknown and can only be approximated via sampling. Extensive experiments on benchmark datasets demonstrate that ROI-Greedy significantly outperforms competing methods in terms of the tradeoff between efficiency and solution quality. PVLDB Reference Format: Tianyuan Jin, Yu Yang, Renchi Yang, Jieming Shi, Keke Huang, and Xiaokui Xiao. Unconstrained Submodular Maximization with Modular Costs: Tight Approximation and Application to Profit Maximization. PVLDB, 14(10): 1756-1768, 2021. doi:10.14778/3467861.3467866