On the Average-Case Performance of Greedy for Maximum Coverage

TL;DR

This paper analyzes the average-case performance of the greedy algorithm for maximum coverage in a specific random model, showing it generally outperforms the worst-case guarantee and identifying conditions for near-perfect approximation.

Contribution

It introduces a new analysis of the greedy algorithm's expected approximation ratio in a random model, with novel analytical tools and insights into its performance regimes.

Findings

Expected approximation ratio exceeds 1-1/e in the random model.

Two simple conditions ensure the ratio is close to 1 for large graphs.

In some regimes, the ratio does not surpass 0.94 on average.

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 Erd\H{o}s-R\'enyi graphs, which may be of independent interest for other random models.