Streaming algorithms for monotone non-submodular function maximization under a knapsack constraint on the integer lattice

• Published in: Theoretical computer science Vol. 937; pp. 39 - 49
• Main Authors: Tan, Jingjing, Wang, Fengmin, Ye, Weina, Zhang, Xiaoqing, Zhou, Yang
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 18.11.2022
• Subjects: Integer lattice; Knapsack constraint; Non-submodular; Streaming algorithms; Streaming algorithms; Integer lattice; Non-submodular; Knapsack constraint
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2022.09.028

The study of non-submodular maximization on the integer lattice is an important extension of submodular optimization. In this paper, streaming algorithms for maximizing non-negative monotone non-submodular functions with knapsack constraint on integer lattice are considered. We first design a two-pass StreamingKnapsack algorithm combining with BinarySearch as a subroutine for this problem. By introducing the DR ratio γ and the weak DR ratio γw of the non-submodular objective function, we obtain that the approximation ratio is min⁡{γ2(1−ε)/2γ+1,1−1/γw2γ−ε}, the total memory complexity is O(Klog⁡K/ε), and the total query complexity for each element is O(log⁡Klog⁡(K/ε2)/ε). Then, we design a one-pass streaming algorithm by dynamically updating the maximal function value among unit vectors along with the currently arriving element. Finally, in order to decrease the memory complexity, we design an improved StreamingKnapsack algorithm and reduce the memory complexity to O(K/ε2). •A two-pass streaming algorithm is proposed for maximizing monotone non-submodular functions with knapsack constraint on integer lattice.•By introducing DR ratio γd (weak DR ratio γw) we obtain the approximation ratio as min⁡{γ2(1−ε)/2γ+1,1−1/γw2γ−ε} with memory O(Klog⁡K/ε).•A one-pass streaming algorithm is designed by dynamically updating the maximal function value among unit vectors.•We design an improve StreamingKnapsack algorithm and reduce the memory complexity to O(K/ε2).