Maximizing the Differences Between a Monotone DR-Submodular Function and a Linear Function on the Integer Lattice

In this paper, we investigate the maximization of the differences between a non-negative monotone diminishing return submodular (DR-submodular) function and a nonnegative linear function on the integer lattice. As it is almost unapproximable for maximizing a submodular function without the condition of nonnegative, we provide weak (bifactor) approximation algorithms for this problem in two online settings, respectively. For the unconstrained online model, we combine the ideas of single-threshold greedy, binary search and function scaling to give an efficient algorithm with a 1/2 weak approximation ratio. For the online streaming model subject to a cardinality constraint, we provide a one-pass (3 - root 5 )/2 weak approximation ratio streaming algorithm. Its memory complexity is (k log k/epsilon), and the update time for per element is (log(2) k/epsilon).