The research area is witnessing significant advancements in the optimization of submodular functions, particularly in scenarios where these functions are approximately submodular or subject to parallel processing constraints. Recent studies have focused on developing efficient algorithms for maximizing approximately submodular functions under cardinality constraints, addressing the inherent challenges posed by errors in function evaluation. These algorithms aim to balance query complexity with approximation guarantees, offering solutions that are both theoretically sound and practically applicable. Additionally, there is a growing interest in applying these optimization techniques to procurement auctions, where the goal is to design mechanisms that are not only computationally efficient but also incentive compatible and individually rational for sellers. This work not only enhances the theoretical understanding of submodular optimization but also opens up new avenues for practical applications in various domains, such as auction design and resource allocation. Notably, the integration of submodular optimization with parallel computing models is proving to be a powerful approach for tackling large-scale problems, with recent algorithms demonstrating significant improvements in both approximation guarantees and computational efficiency.