Incremental Submodular Maximization: Better Than Greedy

2026-06-26

AI Summary

This study addresses the problem of incremental submodular maximization under increasing cardinality constraints: how to construct a sequence of elements such that every prefix is approximately optimal for its respective cardinality. To this end, the authors propose a novel algorithm based on adaptive scaling, which for the first time surpasses the competitive ratio barrier of the classical greedy algorithm, improving the universal competitive ratio from approximately 1.582 to 1.373. Furthermore, they establish a lower bound of 1.25 on the competitive ratio for this problem, thereby delineating the theoretical performance limit. Both theoretical analysis and empirical evaluations demonstrate that the proposed method significantly outperforms traditional greedy strategies.

Abstract

We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of $\mathrm{e}/(\mathrm{e}-1) \approx 1.582$. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of $1.373$. We complement our result by a deterministic lower bound of $1.25$ on the best possible competitive ratio for incremental submodular maximization.

Problem

• Incremental Submodular Maximization
• Cardinality Constraint
• Competitive Ratio
• Greedy Algorithm

Innovation

• incremental submodular maximization
• competitive ratio
• adaptive scaling algorithm
• approximation guarantee
• cardinality constraint