Approximating BP Maximization with Distorted-Based Strategy

Ruiqi Yang, Suixiang Gao, Lu Han, Gaidi Li, Zhongrui Zhao

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