Integer Polyhedra and Combinatorial Optimization

Combinatorial optimization lies at the intersection of discrete mathematics and algorithmic problem solving, providing robust frameworks for tackling complex decision-making challenges. This course offers a clear, structured path through the mathematical foundations of integer polyhedra, duality theory, and exact formulations for network and matching problems. You will gain a rigorous conceptual understanding of how continuous linear programming techniques are applied to guarantee exact integer solutions. What you will learn: Understand core terminology and geometric properties of convex polyhedra and integer hulls; Apply linear programming duality and complementary slackness conditions to verify optimality; Analyze total unimodularity and total dual integrality to identify when linear relaxations yield integer vertices; Formulate combinatorial problems such as shortest path and bipartite matching as linear programs; Examine matching polyhedra in general graphs and the theoretical mechanics behind exact polyhedral descriptions; Modernize optimization models using standard mathematical modeling notation and solver-ready formulations. The course begins with foundational definitions and fundamental theorems of linear programming before guiding you step by step through duality, integrality proofs, and graph optimization models. Designed for learners, computer science students, and analysts new to polyhedral combinatorics, this course requires only basic algebra and curiosity about discrete mathematics. Start reading today to master the structural foundations of combinatorial optimization.