Maximizing Monotone Submodular Functions over the Integer Lattice

TL;DR

This paper develops polynomial-time approximation algorithms for maximizing monotone submodular functions over the integer lattice under various constraints, extending classical results to a more general domain.

Contribution

It introduces the first polynomial-time algorithms achieving near-optimal approximation ratios for submodular maximization over the integer lattice, including cases with and without the diminishing return property.

Findings

Achieves a (1-1/e-ε)-approximation for cardinality, polymatroid, and knapsack constraints.

Provides a faster (1-1/e-ε)-approximation algorithm without requiring diminishing return.

Extends submodular maximization techniques to functions over the integer lattice domain.

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 $f:\mathbb{Z}_+^n \to \mathbb{R}_+$ 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 $(1-1/e-\epsilon)$-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a $(1-1/e-\epsilon)$-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.