Maximizing a Monotone Submodular Function with a Bounded Curvature under a Knapsack Constraint

TL;DR

This paper introduces a polynomial-time algorithm for maximizing monotone submodular functions with bounded curvature under a knapsack constraint, achieving a near-optimal approximation ratio that improves upon previous bounds.

Contribution

It presents the first curvature-aware approximation algorithm for knapsack-constrained submodular maximization, surpassing the classic $1-1/e$ bound.

Findings

Achieves approximation ratio of $1 - c/e - psilon$ for any fixed psilon > 0.

Proves the approximation ratio is tight up to psilon for functions with curvature c.

Provides an improved algorithm for the budget allocation problem.

Abstract

We consider the problem of maximizing a monotone submodular function under a knapsack constraint. We show that, for any fixed $\epsilon > 0$, there exists a polynomial-time algorithm with an approximation ratio $1-c/e-\epsilon$, where $c \in [0,1]$ is the (total) curvature of the input function. This approximation ratio is tight up to $\epsilon$ for any $c \in [0,1]$. To the best of our knowledge, this is the first result for a knapsack constraint that incorporates the curvature to obtain an approximation ratio better than $1-1/e$, which is tight for general submodular functions. As an application of our result, we present a polynomial-time algorithm for the budget allocation problem with an improved approximation ratio.