Randomized Algorithms for Submodular Function Maximization with a $k$-System Constraint

Submodular optimization has numerous applications such as crowdsourcing and viral marketing. In this paper, we study the problem of non-negative submodular function maximization subject to a $k$-system constraint, which generalizes many other important constraints in submodular optimization such as cardinality constraint, matroid constraint, and $k$-extendible system constraint. The existing approaches for this problem are all based on deterministic algorithmic frameworks, and the best approximation ratio achieved by these algorithms (for a general submodular function) is $k+2\sqrt{k+2}+3$. We propose a randomized algorithm with an improved approximation ratio of $(1+\sqrt{k})^2$, while achieving nearly-linear time complexity significantly lower than that of the state-of-the-art algorithm. We also show that our algorithm can be further generalized to address a stochastic case where the elements can be adaptively selected, and propose an approximation ratio of $(1+\sqrt{k+1})^2$ for the adaptive optimization case. The empirical performance of our algorithms is extensively evaluated in several applications related to data mining and social computing, and the experimental results demonstrate the superiorities of our algorithms in terms of both utility and efficiency.