Keywords: [ Deep Learning Theory ]
Given a set V, the problem of unconstrained submodular maximization with modular costs (USM-MC) asks for a subset S \subseteq 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 non-negative 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 networks.
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) - c(S)\ln(f(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+\epsilon)(f(S) - c(S) - c(S) \ln(f(S)/c(S*)), for any \epsilon > 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 trade-off between efficiency and solution quality.