Improved algorithms for non-submodular function maximization problem - Liu, Zhicheng; Jin, Jing; Chang, Hong; Du, Donglei; Zhang, Xiaoyan (Theoretical computer science, 2022)
•We combine the distorted objective function, the threshold and the greedy algorithms.•This problem includes some previously studied problems as special cases, such as submodular+supermodular maximization, γ-weakly submodular function maximization.•The approximation ratio in Algorithm 1 is the minimum of 1−kfe−1 and 1−(kg)2, improving the result obtained by Bai et al.•Algorithm 1 has an approximation ratio 1−(1−γ+γα)e−γ, improving the result obtained by Bian et al.•Set g(S)=0. Algorithm 2 has an approximation ratio γγ+1, improving the approximate ratio 1−12γ by Wang et al. The concept of submodularity finds wide applications in data science, artificial intelligence, and machine learning, providing a boost to the investigation of new ideas, innovative techniques, and creative algorithms to solve different submodular optimization problems arising from a diversity of applications. However pure submodular or supermodular problems only represent a small portion of the problems we are facing in real life applications. The main focus of this work is to consider a non-submodular function maximization problem subject to a cardinality constraint, where the objective function is the sum of a monotone γ-weakly submodular function and a supermodular function. This problem includes some previously studied problems as special cases, such as the submodular+supermodular maximization problem when γ=1, and the γ-weakly submodular function maximization problem when the supermodular function is void. We present greedy algorithms for this generalized problem under both offline and streaming models, improving existing results.