Approximating BP Maximization with Distorted-Based Strategy

• Published in: Parallel and Distributed Computing, Applications and Technologies Vol. 13148; pp. 452 - 459
• Main Authors: Yang, Ruiqi, Gao, Suixiang, Han, Lu, Li, Gaidi, Zhao, Zhongrui
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2022; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Approximation algorithms; Distorted threshold; Fairness; Stream model; Submodular maximization
• ISBN: 9783030967710; 3030967719
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-96772-7_41

We study a problem of maximizing the sum of a suBmodular and suPermodular (BP) function, denoted as maxS⊆V,|S|≤kG(S)+L(S) $$\max _{\mathcal {S}\subseteq \mathcal {V}, |\mathcal {S}|\le k}\mathcal {G}(\mathcal {S})+\mathcal {L}(\mathcal {S})$$ , where G(·) $$\mathcal {G}(\cdot )$$ is non-negative monotonic and submodular, L(·) $$\mathcal {L}(\cdot )$$ is monotonic and supermodular. In this paper, we consider the K $$\mathcal {K}$$ -cardinality constrained BP maximization under a streaming setting. Denote κ $$\kappa $$ as the supermodular curvature of L $$\mathcal {L}$$ . Utilizing a distorted threshold-based technique, we present a first (1-κ)/(2-κ) $$(1-\kappa )/(2-\kappa )$$ -approximation semi-streaming algorithm and then implement it by lazily guessing the optimum threshold and yield a one pass, O(ε-1log((2-κ)K/(1-κ)2)) $$\mathcal {O}(\varepsilon ^{-1}\log ((2-\kappa )\mathcal {K}/(1-\kappa )^2))$$ memory complexity, ((1-κ)/(2-κ)-O(ε)) $$((1-\kappa )/(2-\kappa )-\mathcal {O}(\varepsilon ))$$ -approximation. We further study the BP maximization with fairness constrains and develop a distorted greedy-based algorithm, which gets a (1-κ)/(2-κ) $$(1-\kappa )/(2-\kappa )$$ -approximation for the extended fair BP maximization.