Randomized Algorithms for Monotone Submodular Function Maximization on the Integer Lattice

TL;DR

This paper introduces a randomized algorithm for maximizing monotone DR-submodular functions on the integer lattice with approximation guarantees, demonstrating improved efficiency over existing methods in synthetic scenarios.

Contribution

It extends stochastic greedy algorithms to the integer lattice domain for DR-submodular maximization, providing probabilistic approximation guarantees.

Findings

The algorithm achieves an approximation ratio of O(1 - 1/e - epsilon).

It outperforms existing methods in synthetic experiments in terms of speed.

The approach generalizes set submodular maximization techniques to the integer lattice.

Abstract

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.