Streaming submodular maximization under d-knapsack constraints

Submodular optimization is a key topic in combinatorial optimization, which has attracted extensive attention in the past few years. Among the known results, most of the submodular functions are defined on set. But recently some progress has been made on the integer lattice. In this paper, we study two problem of maximizing submodular functions with d-knapsack constraints. First, for the problem of maximizing DR-submodular functions with d-knapsack constraints on the integer lattice, we propose a one pass streaming algorithm that achieves a $$\frac{1-\theta }{1+d}$$ 1 - θ 1 + d -approximation with $$O\left( \frac{\log (d\beta ^{-1})}{\beta \epsilon }\right) $$ O log ( d β - 1 ) β ϵ memory complexity and $$O\left( \frac{\log (d\beta ^{-1})}{\epsilon }\log \Vert {\textbf {b}} \Vert _{\infty }\right) $$ O log ( d β - 1 ) ϵ log ‖ b ‖ ∞ update time per element, where $$\theta =\min (\alpha +\epsilon , 0.5+\epsilon )$$ θ = min ( α + ϵ , 0.5 + ϵ ) and $$\alpha , \beta $$ α , β are the upper and lower bounds for the cost of each item in the stream. Then we devise an improved streaming algorithm to reduce the memory complexity to $$O(\frac{d}{\beta \epsilon })$$ O ( d β ϵ ) with unchanged approximation ratio and query complexity. Then for the problem of maximizing submodular functions with d-knapsack constraints under noise, we design a one pass streaming algorithm. When $$\varepsilon \rightarrow 0$$ ε → 0 , it achieves a $$\frac{1}{1-\alpha +d}$$ 1 1 - α + d -approximate ratio, memory complexity $$O\left( \frac{\log (d\beta ^{-1})}{\beta \epsilon }\right) $$ O log ( d β - 1 ) β ϵ and query complexity $$O\left( \frac{\log (d\beta ^{-1})}{\epsilon }\right) $$ O log ( d β - 1 ) ϵ per element. As far as we know, these two are the first streaming algorithms under their corresponding problems.