Typical Approximation Performance for Maximum Coverage Problem

This study investigated typical performance of approximation algorithms known as belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problems on sparse biregular random graphs. After using the cavity method for a corresponding hard-core lattice--gas model, results show that two distinct thresholds of replica-symmetry and its breaking exist in the typical performance threshold of belief propagation. In the low-density region, the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses. Although the greedy algorithm and linear-programming relaxation have the same approximation ratio in worst-case performance, their typical performance thresholds are mutually different, indicating the importance of typical performance. Results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms.