Combinatorial Optimization: Matroids and Submodular Functions

Combinatorial optimization provides the mathematical backbone for solving complex discrete decision problems in computer science and operations research. This text-based course guides you through core theoretical models and structural properties, breaking down abstract concepts into approachable, step-by-step principles. You will build a solid intuition for linear and integer programming formulations, understand the structural beauty of matroids, and examine how polynomial-time algorithms solve challenging discrete problems. What you'll learn: - Understand foundational definitions of independent systems, matroid bases, and rank functions - Analyze submodular function properties and approaches to submodular minimization - Explore matroid intersection theorems and their algorithmic significance - Connect discrete optimization to polyhedral combinatorics and total dual integrality - Apply min-max relations and classical results like the König-Egerváry theorem - Formulate combinatorial graph problems using modern linear programming relaxations The course begins with fundamental definitions and graph-theoretic prerequisites before advancing into structural theorems, duality proofs, and polyhedral analysis through clear text explanations and worked mathematical examples. This course is designed for beginners in advanced discrete mathematics, computer science students, and programmers seeking a rigorous introduction to combinatorial optimization with no specialized background required beyond basic linear algebra and discrete math basics. Start reading today to master the mathematical foundations of combinatorial optimization.