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 nonnegative 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-√5)/2 weak approximation ratio streaming algorithm.Its memory complexity is(k log k/ε),and the update time for per element is(log^(2)k/ε).