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In this work, we study the classic submodular maximization problem underknapsack constraints and beyond. We first present an$(7/16-\varepsilon)$-approximate algorithm for single knapsack constraint,which requires $O(n\cdot\max\{\varepsilon^{-1},\log\log n\})$ queries, and twopasses in the streaming setting. This provides improvements in approximationratio, query complexity and number of passes on the stream. We next show thatthere exists an $(1/2-\varepsilon)$-approximate deterministic algorithm forconstant number of binary packing constraints, which achieves a querycomplexity of $O_{\varepsilon}(n\cdot\log \log n)$. One salient feature of ourdeterministic algorithm is, both its approximation ratio and time complexityare independent of the number of constraints. Lastly we present nearly lineartime algorithms for the intersection of $p$-system and $d$ knapsack constraint,we achieve approximation ratio of $(1/(p+\frac{7}{4}d+1)-\varepsilon)$ formonotone objective and $(p/(p+1)(2p+\frac{7}{4}d+1)-\varepsilon)$ fornon-monotone objective.

Nearly Linear Time Deterministic Algorithms for Submodular Maximization Under Knapsack Constraint and Beyond