Regularized two-stage submodular maximization under streaming

• Published in: Science China. Information sciences Vol. 65; no. 4; p. 140602
• Main Authors: Yang, Ruiqi, Xu, Dachuan, Guo, Longkun, Zhang, Dongmei
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.04.2022; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computer Science; Information Systems and Communication Service; Mathematical analysis; Maximization; Optimization; Research Paper; threshold technique; submodular maximization; two-stage; approximation algorithms; streaming model
• ISSN: 1674-733X; 1869-1919
• DOI: 10.1007/s11432-020-3420-9

In the problem of maximizing regularized two-stage submodular functions in streams, we assemble a family ℱ of m functions each of which is submodular and is visited in a streaming style that an element is visited for only once. The aim is to choose a subset S of size at most ℓ from the element stream V , so as to maximize the average maximum value of these functions restricted on S with a regularized modular term. The problem can be formally casted as max S ⊆ V , | S | ⩽ ℓ 1 m ∑ i = 1 m max T ⊆ S , | T | ⩽ k [ f i ( T ) − c ( T ) ] , where c : V → ℝ + is a non-negative modular function and f i : 2 V → ℝ + , ∀ i ∈ { 1 , … , m } is a non-negative monotone non-decreasing submodular function. The well-studied regularized problem of max S ⊆ V , | S | ⩽ k f ( S ) − c ( S ) is exactly a special case of the above regularized two-stage submodular maximization by setting m = 1 and ℓ = k . Although f (·) − c (·) is submodular, it is potentially negative and non-monotone and admits no constant multiplicative factor approximation. Therefore, we adopt a slightly weaker notion of approximation which constructs S such that f ( S ) − c ( S ) ⩾ ρ · f ( O ) − c ( O ) holds against optimum solution O for some ρ ∈ (0, 1). Eventually, we devise a streaming algorithm by employing the distorted threshold technique, achieving a weaker approximation ratio with ρ = 0.2996 for the discussed regularized two-stage model.