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

Abstract

In this work, we study the classic submodular maximization problem under knapsack 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 two passes in the streaming setting. This provides improvements in approximation ratio, query complexity and number of passes on the stream. We next show that there exists an $(1/2-\varepsilon)$-approximate deterministic algorithm for constant number of binary packing constraints, which achieves a query complexity of $O_{\varepsilon}(n\cdot\log \log n)$. One salient feature of our deterministic algorithm is, both its approximation ratio and time complexity are independent of the number of constraints. Lastly we present nearly linear time algorithms for the intersection of $p$-system and $d$ knapsack constraint, we achieve approximation ratio of $(1/(p+\frac{7}{4}d+1)-\varepsilon)$ for monotone objective and $(p/(p+1)(2p+\frac{7}{4}d+1)-\varepsilon)$ for non-monotone objective.