A Nearly-linear Time Algorithm for Submodular Maximization with a Knapsack Constraint

TL;DR

This paper introduces a nearly-linear time algorithm for maximizing monotone submodular functions under a knapsack constraint, achieving near-optimal approximation with improved theoretical efficiency.

Contribution

It presents a novel algorithm that overcomes the quadratic time bottleneck of multilinear extension methods, maintaining a sparse fractional solution for efficiency.

Findings

Achieves a $1 - 1/e - \epsilon$ approximation ratio.

Uses significantly fewer function evaluations than previous methods.

Provides a theoretically interesting approach despite being impractical.

Abstract

We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, $1 - 1/e - \epsilon$ approximation, using $(1/\epsilon)^{O(1/\epsilon^4)} n \log^2{n}$ function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to $\Omega(n^2)$ running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.