On the Average-Case Performance of Greedy for Maximum Coverage

2026-04-27

AI Summary

This study addresses the gap between the worst-case theoretical guarantee of the greedy algorithm for the maximum coverage problem—namely, the well-known approximation ratio of \(1 - 1/e\)—and its typically superior empirical performance. By analyzing typical instances under a left-regular random model and employing techniques from differential equations and random graph theory, the work establishes the first theoretical characterization of the algorithm’s expected approximation ratio in average-case settings. The main contributions include proving that the expected approximation ratio strictly exceeds \(1 - 1/e\) for all parameter regimes, identifying conditions under which the ratio approaches 1 in large-scale graphs, and demonstrating the existence of parameter intervals where the expected ratio does not surpass 0.94.

Abstract

For the classical maximum coverage problem, the greedy algorithm achieves a worst-case $1-1/e$ approximation, which is optimal unless $\text{P} = \text{NP}$. The notion of coverage appears in a wide range of optimization tasks, where empirical evaluations indicate approximation ratios close to $1$ for the greedy algorithm on real data. Random models have provided average-case justifications for the empirical performance of many well-known algorithms, but little is known about the average-case performance of greedy for maximum coverage. We analyze the expected approximation ratio of the greedy algorithm in a random model, which we call the left-regular random model. We first show that, for all parameter settings of this model, the expected approximation ratio of the greedy algorithm improves by a constant over its worst-case $1-1/e$ guarantee. We then identify two simple conditions, either of which ensures that the expected approximation ratio is close to $1$ for sufficiently large graphs. Finally, we show that there is a regime where greedy does not achieve an expected approximation better than $0.94$. To obtain these results, we develop analytical tools, including a novel application of the differential equation method and a connection to maximum matching in Erdős-Rényi graphs, which may be of independent interest for other random models.

Problem

• maximum coverage
• greedy algorithm
• average-case performance
• approximation ratio
• random model

Innovation

• maximum coverage
• greedy algorithm
• average-case analysis
• differential equation method
• random graph model