Approximating BP Maximization with Distorted-Based Strategy - Yang, Ruiqi; Gao, Suixiang; Han, Lu; Li, Gaidi; Zhao, Zhongrui (Parallel and Distributed Computing, Applications and Technologies, 2022)
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(·) $$\mathcal {G}(· )$$ is non-negative monotonic and submodular, L(·) $$\mathcal {L}(· )$$ is monotonic and supermodular. In this paper, we consider the K $$\mathcal {K}$$ -cardinality constrained BP maximization under a streaming setting. Denote κ $$κ $$ as the supermodular curvature of L $$\mathcal {L}$$ . Utilizing a distorted threshold-based technique, we present a first (1-κ)/(2-κ) $$(1-κ )/(2-κ )$$ -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)) $$\mathcal {O}(ε ^{-1}\log ((2-κ )\mathcal {K}/(1-κ )^2))$$ memory complexity, ((1-κ)/(2-κ)-O(ε)) $$((1-κ )/(2-κ )-\mathcal {O}(ε ))$$ -approximation. We further study the BP maximization with fairness constrains and develop a distorted greedy-based algorithm, which gets a (1-κ)/(2-κ) $$(1-κ )/(2-κ )$$ -approximation for the extended fair BP maximization.