Approximation Algorithms and Semidefinite Programming for Max-Cut

Combinatorial optimization problems like Max-Cut appear across computer science, network design, and data science, yet solving them exactly is often computationally intractable. This text-based course guides you through foundational approximation techniques and mathematical optimization strategies used to tackle these challenging problems efficiently. You will begin with core graph theory concepts before exploring quadratic integer programming formulations and powerful semidefinite programming (SDP) relaxations. Through structured written lessons and code snippets, you will discover how modern approximation algorithms balance theoretical guarantees with practical computational performance. What you'll learn: - Understand graph partitioning fundamentals and the formal definition of the Max-Cut problem - Learn randomized approximation techniques and analyze simple 2-approximation baselines - Master quadratic integer programming representations and vector program transformations - Apply semidefinite programming relaxations to convert discrete optimization into solvable convex problems - Explore the Goemans-Williamson randomized hyperplane rounding method and its approximation ratio analysis - Practice modeling and solving semidefinite programs using modern Python optimization libraries The course begins with foundational terminology, linear algebra essentials, and graph definitions before advancing step-by-step through mathematical modeling, SDP solving, and rounding techniques. It is designed for learners with basic programming knowledge and an interest in algorithms, requiring no prior experience with semidefinite optimization. Start reading today to expand your toolkit with advanced algorithmic design and approximation methods.