Deterministic streaming algorithms for non-monotone submodular maximization

Submodular maximization is a significant area of interest in combinatorial optimization. It has various real-world applications. In recent years, streaming algorithms for submodular maximization have gained attention, allowing realtime processing of large data sets by examining each piece of data only once. However, most of the current state-of-the-art algorithms are only applicable to monotone submodular maximization. There are still significant gaps in the approximation ratios between monotone and non-monotone objective functions.

In this paper, we propose a streaming algorithm framework for non-monotone submodular maximization and use this framework to design deterministic streaming algorithms for the d-knapsack constraint and the knapsack constraint. Our 1-pass streaming algorithm for the d-knapsack constraint has a \({1 \over 4(d+1)}-\epsilon\) $$\frac{1}{4(d+1)}-ϵ$$ approximation ratio, using \({\cal O}\left({{\tilde B}\log{\tilde B} \over \epsilon}\right)\) $$\mathcal{O}\left(\frac{\tilde{B}\log \tilde{B}}{ϵ}\right)$$ memory, and \({\cal O}\left({\log{\tilde B} \over \epsilon}\right)\) $$\mathcal{O}\left(\frac{\log \tilde{B}}{ϵ}\right)$$ query time per element, where \({\tilde B}=\min(n,b)\) $$\tilde{B}=min(n,b)$$ is the maximum number of elements that the knapsack can store. As a special case of the d-knapsack constraint, we have the 1-pass streaming algorithm with a 1/8 − ϵ approximation ratio to the knapsack constraint. To our knowledge, there is currently no streaming algorithm for this constraint when the objective function is non-monotone, even when d = 1. In addition, we propose a multi-pass streaming algorithm with 1/6 − ϵ approximation, which stores \({\cal O}({\tilde B})\) $$\mathcal{O}(\tilde{B})$$ elements.