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On the Average-Case Performance of Greedy for Maximum Coverage

Authors Eric Balkanski , Jason Chatzitheodorou , Flore Sentenac

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Maximum Coverage
  • Greedy Algorithm
  • Average-Case Analysis
  • Differential Equation Method
  • Random Graphs

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Abstract

For the classical maximum coverage problem, the greedy algorithm achieves a worst-case 1-1/e approximation, which is optimal unless P = 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.

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Eric Balkanski, Jason Chatzitheodorou, and Flore Sentenac. On the Average-Case Performance of Greedy for Maximum Coverage. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.20

Author Details

Eric Balkanski
  • Columbia IEOR, New York, NY, USA
Jason Chatzitheodorou
  • Columbia IEOR, New York, NY, USA
Flore Sentenac
  • HEC, Paris, France

Funding

Eric Balkanski and Jason Chatzitheodorou were supported in part by NSF grants CCF-2210501 and IIS-2147361.

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