A Streaming Model for Monotone Lattice Submodular Maximization with a Cardinality Constraint

Abstract

In this paper, we consider a streaming model of maximizing monotone lattice submodular function with a cardinality constraint on the integer lattice. As (lattice) submodularity does not imply the diminishing return property on the integer lattice, we introduce the Sieve-Streaming algorithm combining with a modified binary search subroutine to solve the problem. We also show it is with an approximation ratio \(1/2-\epsilon \), a memory complexity \(O( \epsilon ^{-1} k\log k)\), and a query complexity \(O( \epsilon ^{-2}\log ^2 k )\) per element.