Streaming Algorithms for Maximizing DR-Submodular Functions with d-Knapsack Constraints

Bin Liu, Zihan Chen, Hongmin W. Du

Abstract: The problem of maximizing submodular functions has received considerable attention in the last few years. However, 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 streaming algorithms for the problem of maximizing DR-submodular functions with d-knapsack constraints on the integer lattice. We first propose a one pass streaming algorithm that achieves a $$\frac{1-\theta }{1+d}$$ -approximation with $$O(\frac{\log (d\beta ^{-1})}{\beta \epsilon })$$ memory complexity and $$O(\frac{\log (d\beta ^{-1})}{\epsilon }\log \Vert {{\boldsymbol{b}}}\Vert _{\infty })$$ update time per element, where $$\theta =\min (\alpha +\epsilon , 0.5+\epsilon )$$ and $$\alpha , \beta $$ 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 $$O(\frac{d}{\beta \epsilon })$$ with unchanged approximation ratio and query complexity. As far as we know, this is the first streaming algorithm on the integer lattice under this constraint.