Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

TL;DR

This paper improves the approximation guarantee of the modified greedy algorithm for monotone submodular maximization with a knapsack constraint from 0.357 to 0.405, clarifies a long-standing misconception, and introduces a data-dependent upper bound to enhance algorithm efficiency.

Contribution

The paper provides a tighter approximation factor for the modified greedy algorithm and clarifies previous theoretical misconceptions, along with a data-dependent bound for better practical performance.

Findings

Achieves an approximation factor of 0.405, surpassing previous bounds.

Closes a gap in the proof of the approximation factor 0.393.

Demonstrates the tightness of the upper bound with real-world data.

Abstract

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of $0.405$, which significantly improves the known factors of $0.357$ given by Wolsey and $(1-1/\mathrm{e})/2\approx 0.316$ given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of $(1-1/\sqrt{\mathrm{e}})\approx 0.393$ in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than $0.405$ between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.