Explore advanced concepts in Network Design Problems through this lecture on approximation algorithms for hard augmentation problems. Delve into the intricacies of increasing graph edge-connectivity, starting with the fundamental Minimum Spanning Tree Problem and progressing to more complex scenarios. Examine recent approaches and advancements in the field of Connectivity Augmentation, including both unweighted and weighted Tree Augmentation. Learn how classical Combinatorial Optimization techniques are combined with innovative ideas to tackle these challenging problems. Gain insights into the latest research and methodologies used to solve hard augmentation problems in graph theory and network design.