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Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. Streaming algorithms for submodule maximization have gained attention in recent years, allowing for real-time processing of large data sets by looking at each piece of data only once. However, most of the state-of-the-art algorithms are subject to monotone cardinality constraint. There remain non-negligible gaps with respect to approximation ratios between cardinality and other constraints like $d$-knapsack in non-monotone submodular maximization. In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the cardinality constraint, we provide a deterministic $1/6-\epsilon$ approximation algorithm with $O(\frac{k\log k}{\epsilon})$ memory and sublinear query time, while the previous best algorithm is randomized with a $0.1921$ approximation ratio. To the best of our knowledge, this is the first deterministic streaming algorithm for the cardinality constraint. For the $d$-knapsack constraint, we provide a deterministic $\frac{1}{4(d+1)}-\epsilon$ approximation algorithm with $O(\frac{b\log b}{\epsilon})$ memory and $O(\frac{\log b}{\epsilon})$ query time per element. To the best of our knowledge, there is currently no streaming algorithm for this constraint. |
