Streaming Algorithms For Maximizing Monotone Dr-Submodular Functions With A Cardinality Constraint On The Integer Lattice

Emerging applications such as optimal budget allocation and sensor placement impose problems of maximizing variants of submodular functions with constraints under a streaming setting. In this paper, we first devise a streaming algorithm based on Sieve-Streaming for maximizing a monotone diminishing return submodular (DR-submodular) function with a cardinality constraint on the integer lattice and show it is a one-pass algorithm with approximation ratio 1/2 - epsilon. The key idea to ensure one pass for the algorithm is to combine binary search for determining the level of an element with the exponential-growth method for estimating the OPT. Inspired by Sieve-Streaming++, we then improve the memory of the algorithm to O(k/epsilon) and the query complexity to O(k log(2) k/epsilon).