Maximizing Monotone Submodular Functions over the Integer Lattice

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function f : Z(+)(n) -> R+ is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of the submodularity when the domain is the integer lattice. Then, we show polynomial-time (1 - 1/e - epsilon)-approximation algorithm for cardinality constraints, polymatroid constraints, and knapsack constraints. For a cardinality constraint, we also show a (1 - 1/e - epsilon)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property. Our algorithms for a polymatroid constraint and a knapsack constraint first extend the domain of the objective function to the Euclidean space and then run the continuous greedy algorithm. We give two different kinds of continuous extensions, one is for polymatroid constraints and the other is for knapsack constraints, which might be of independent interest.