Metric Geometry for Combinatorial Optimization

Combinatorial optimization problems often present computational hurdles that exact algorithms cannot solve efficiently. This text-based course introduces how metric geometry provides powerful mathematical tools to construct approximation algorithms with provable performance guarantees. You will explore the foundational principles linking finite metric spaces, geometric embeddings, and algorithmic design. Through clear written explanations and analytical problem walkthroughs, you will develop the ability to formulate optimization problems geometrically and analyze approximation factors. What you'll learn: - Understand core metric geometry concepts, including metric spaces, distortion, and embedding techniques - Apply Bourgain's embedding theorem and low-distortion mappings into normed spaces - Formulate linear programming and semidefinite programming (SDP) relaxations for NP-hard graph problems - Design approximation algorithms for fundamental graph cuts, including Sparsest Cut and Balanced Cut - Analyze rounding methods and distance preservation in metric representations - Solve structural layout problems such as Minimum Linear Arrangement using geometric bounds The course begins with fundamental definitions of metric spaces and algorithmic complexity before progressing through embedding theorems, convex relaxations, and cut approximation strategies. Designed for beginners in theoretical computer science and discrete mathematics, this course requires only an introductory familiarity with graph concepts and linear algebra. Start reading today to master geometric methods in modern algorithm design.