Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint

Arising from practical problems such as in sensor placement and influence maximization in social network, submodular and non-submodular maximization on the integer lattice has attracted much attention recently. In this work, we consider the problem of maximizing the sum of a monotone non-negative diminishing return submodular (DR-submodular) function and a supermodular function on the integer lattice subject to a cardinality constraint. By exploiting the special combinatorial structures in the problem, we introduce a decreasing threshold greedy algorithm with a binary search as its subroutine to solve the problem. To avoid introducing the diminishing return ratio and submodularity ratio of the objective function, we generalize the total curvatures of submodular functions and supermodular functions to the integer lattice version. We show that our algorithm has a constant approximation ratio parameterized by the new introduced total curvatures on integer lattice with a polynomial query complexity.