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Explore cutting-edge approximation algorithms for classic network design problems in this comprehensive lecture. Delve into the Prize-collecting Framework, which addresses optimization challenges where multiple demands seek service from the most cost-effective structure. Learn about innovative approaches to problems such as Minimum Spanning Tree, Steiner Tree, TSP, Steiner Forest, and Steiner Networks. Discover how modern techniques, including clustering algorithms and recursive methodologies, are applied to solve these complex issues. Gain insights into the latest advancements in approximation algorithms for Prize-collecting Steiner Forest and Prize-collecting Steiner Tree, presented by Mohammad Taghi Hajiaghayi at ICBS2024.