Abstract 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 submodularity when the domain is the integer lattice. Given this, we design polynomial-time (1 − 1/e − ǫ)-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a (1 − 1/e − ǫ)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.
On integer lattice, Soma et al. REF studied the problem of maximizing monotone DR-submodular exhaustively, where they designed algorithms with (1 − 1/e − ε)-approximation under the cardinality, polymatroid and knapsack constraint.
Maximizing Monotone Submodular Functions over the Integer Lattice
arXiv: Data Structures and Algorithms
Mathematics Computer Science