  Optimization problems with set submodular objective functions have many
real-world applications. In discrete scenarios, where the same item can be
selected more than once, the domain is generalized from a 2-element set to a
bounded integer lattice. In this work, we consider the problem of maximizing a
monotone submodular function on the bounded integer lattice subject to a
cardinality constraint. In particular, we focus on maximizing DR-submodular
functions, i.e., functions defined on the integer lattice that exhibit the
diminishing returns property. Given any epsilon > 0, we present a randomized
algorithm with probabilistic guarantees of O(1 - 1/e - epsilon) approximation,
using a framework inspired by a Stochastic Greedy algorithm developed for set
submodular functions by Mirzasoleiman et al. We then show that, on synthetic
DR-submodular functions, applying our proposed algorithm on the integer lattice
is faster than the alternatives, including reducing a target problem to the set
domain and then applying the fastest known set submodular maximization
algorithm.
