Approximating ${\text{(}m}_B\mathrm{,}{m}_P\text{)}$-Monotone BP Maximization and Extensions

The paper proposes the optimization problem of maximizing the sum of suBmodular and suPermodular (BP) functions with partial monotonicity under a streaming fashion. In this model, elements are randomly released from the stream and the utility is encoded by the sum of partial monotone suBmodular and suPermodular functions. The goal is to determine whether a subset from the stream of size bounded by parameter $k$ subject to the summarized utility is as large as possible. In this work, a threshold-based streaming algorithm is presented for the BP maximization that attains a $\mathrm{(}\mathrm{(}\mathrm{1}\mathrm{-}\kappa \mathrm{)}\mathrm{/}\mathrm{(}\mathrm{2}\mathrm{-}\kappa \mathrm{)}\mathrm{-}𝒪\mathsf{}\mathrm{(}\epsilon \mathrm{)}\mathrm{)}$-approximation with $𝒪\mathsf{}\mathrm{(}\mathrm{1}\mathrm{/}{\epsilon }^{\mathrm{4}}\mathsf{}{\log }^{\mathrm{3}}\mathsf{}\mathrm{(}\mathrm{1}\mathrm{/}\epsilon \mathrm{)}\mathsf{}\log \mathsf{}\mathrm{(}\mathrm{(}\mathrm{2}\mathrm{-}\kappa \mathrm{)}\mathsf{}k\mathrm{/}{\mathrm{(}\mathrm{1}\mathrm{-}\kappa \mathrm{)}}^{\mathrm{2}}\mathrm{)}\mathrm{)}$ memory complexity, where $\kappa$ denotes the parameter of supermodularity ratio. We further consider a more general model with fair constraints and present a greedy-based algorithm that obtains the same approximation. We finally investigate this fair model under the streaming fashion and provide a $\mathrm{(}{\mathrm{(}\mathrm{1}\mathrm{-}\kappa \mathrm{)}}^{\mathrm{4}}\mathrm{/}{\mathrm{(}\mathrm{2}\mathrm{-}\mathrm{2}\mathsf{}\kappa \mathrm{+}{\kappa }^{\mathrm{2}}\mathrm{)}}^{\mathrm{2}}\mathrm{-}𝒪\mathsf{}\mathrm{(}\epsilon \mathrm{)}\mathrm{)}$-approximation algorithm.