Improved algorithms for non-submodular function maximization problem

• Published in: Theoretical computer science Vol. 931; pp. 49 - 55
• Main Authors: Liu, Zhicheng, Jin, Jing, Chang, Hong, Du, Donglei, Zhang, Xiaoyan
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 29.09.2022
• Subjects: Cardinality constraint; Greedy algorithm; Non-submodular function maximization; Offline model; Streaming model; Non-submodular function maximization; Greedy algorithm; Streaming model; Offline model; Cardinality constraint
• ISSN: 0304-3975
• DOI: 10.1016/j.tcs.2022.07.029

•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.