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 DRsubmodular functions has many applications in machine learning that cannot be captured by submodular set functions. In this paper, we present a 1 2+ -approximation algorithm with a running time of roughly O( n log 2 B), where n is the size of the ground set, B is the maximum value of a coordinate, and > 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.
Simultaneously, they REF considered non-monotone DR-submodular maximization over the integer lattice, and presented a 1/(2 + ε)-approximate algorithm within polynomial time.
Non-monotone DR-Submodular Function Maximization
ArXiv
ArXiv
Mathematics Computer Science