Non-Monotone DR-Submodular Function Maximization

Tasuku Soma, Yuichi Yoshida

Abstract: We consider non-monotone DR-submodular function maximization, where DR-submodularity (diminishing return submodularity) is an extension of submodularity for functions over the integer lattice based on the concept of the diminishing return property. Maximizing non-monotone DR-submodular functions has many applications in machine learning that cannot be captured by submodular set functions. In this paper, we present a \frac{1}{2+\epsilon}$-approximation algorithm with a running time of roughly O(\frac{n}{\epsilon}\log^2 B), where n$ is the size of the ground set, B$ is the maximum value of a coordinate, and \epsilon > 0 is a parameter. The approximation ratio is almost tight and the dependency of running time on B$ is exponentially smaller than the naive greedy algorithm. Experiments on synthetic and real-world datasets demonstrate that our algorithm outputs almost the best solution compared to other baseline algorithms, whereas its running time is several orders of magnitude faster.