Approximation Algorithms and Primal-Dual Graph Optimization

Network optimization often involves computationally hard problems that cannot be solved efficiently by brute force. Understanding linear programming relaxations and duality provides a systematic framework for designing provably effective approximation algorithms for these challenges. This written course guides you through core theoretical foundations and practical algorithmic strategies. You will build a clear understanding of how linear programs model discrete combinatorial problems, how dual formulations reveal structural insights, and how primal-dual techniques produce bounded approximation guarantees. What you'll learn: - Understand the fundamentals of linear programming relaxations and polyhedral descriptions for graph problems - Apply linear programming duality concepts, complementary slackness, and separation oracles - Analyze vertex-weighted multicut problems using multiflows and path formulations - Master half-integrality properties and derive 2-approximation algorithms - Implement primal-dual schema principles to solve the minimum branching (arborescence) problem - Practice formulating integer programs and their continuous relaxations through structured text exercises The course begins with foundational definitions of graphs, linear inequalities, and duality before progressing through cut problems, multiflows, and primal-dual algorithms. Each concept is presented with clear step-by-step mathematical reasoning and text explanations. This course is designed for learners with an interest in discrete mathematics, computer science theory, or algorithm design. Basic familiarity with standard graph notation and algebra is helpful, but all key concepts are introduced from the ground up. Start reading today to expand your toolkit in mathematical optimization and approximation algorithms.