A Refined Analysis of Submodular Greedy

TL;DR

This paper provides a refined analysis of the greedy algorithm for monotone submodular maximization under a knapsack constraint, improving theoretical bounds and reducing enumeration complexity.

Contribution

It introduces a new analysis that reduces enumeration from size three to two and improves the approximation upper bound for the greedy algorithm.

Findings

Reduced enumeration in the approximation analysis

Improved upper bound of 0.42945 for the greedy algorithm

Enhanced understanding of greedy heuristic performance

Abstract

Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: $(1)$ reduce the enumeration in the tight $(1-e^{-1})$-approximation of [Sviridenko 04] from subsets of size three to two; $(2)$ present an improved upper bound of $0.42945$ for the classic algorithm which returns the better between a single element and the output of the greedy heuristic.