Submodular Optimization and Polymatroids: Foundations and Algorithms

Submodular functions play a central role in discrete mathematics, theoretical computer science, and modern machine learning. This text-based course provides a structured introduction to submodularity, guiding you from fundamental set function properties to advanced polymatroid optimization. You will build a solid mathematical intuition for submodular rank functions and discover how structural properties enable efficient algorithmic solutions. What you'll learn: Understand core definitions and properties of submodular functions on set systems; Explore polymatroids, base polyhedra, and extended polymatroid geometries; Apply greedy algorithms to optimize linear objectives over polymatroids; Analyze total dual integrality and its implications for integer programming; Study foundational algorithms for submodular function minimization and truncation operations; Connect submodular optimization techniques to modern combinatorial and data science applications. The course begins with foundational definitions and mathematical terminology before progressing to geometric representations, duality, and algorithmic procedures. This guide is designed for students, researchers, and software engineers with basic mathematical maturity seeking a clear, self-paced introduction to discrete optimization. Start reading today to deepen your understanding of combinatorial optimization.