This book, "Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties," provides a comprehensive overview of the design and analysis of approximation algorithms for NP-hard combinatorial optimization problems. Authored by Giorgio Ausiello, Alberto Marchetti-Spaccamela, Pierluigi Crescenzi, and Viggo Kann, the book is structured into several chapters that cover fundamental concepts, design techniques, and advanced topics in approximation algorithms. The first chapter introduces the complexity of optimization problems, including the analysis of algorithms, complexity classes, and reducibility among problems. It also discusses the complexity of optimization problems and their classification into PO (Polynomial Optimization), NPO (Non-Deterministic Polynomial Optimization), and NP-hard optimization problems. Subsequent chapters delve into various design techniques for approximation algorithms, such as the greedy method, sequential algorithms for partitioning problems, local search, linear programming-based algorithms, dynamic programming, and randomized algorithms. The book also explores the concept of approximation classes, including APX (Approximable Problems), PTAS ( Polynomial-Time Approximation Schemes), and FPTAS (Fully Polynomial-Time Approximation Schemes). The sixth to eighth chapters focus on theoretical aspects, including the PCP (Probabilistically Checkable Proofs) model, the PCP theorem, and the study of approximability classes. The ninth and tenth chapters extend the discussion to probabilistic analysis of approximation algorithms and heuristic methods, respectively. The book includes a rich compendium of over 200 problems, categorized into twelve subjects, with each entry providing the best and worst approximation results. It also features exercises and an extensive bibliography, making it suitable for both undergraduate and graduate students in computer science and mathematics. The authors acknowledge the contributions of numerous students and friends who helped refine the content and express gratitude to their colleague and friend, Marco Protasi, who passed away before the book's completion.This book, "Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties," provides a comprehensive overview of the design and analysis of approximation algorithms for NP-hard combinatorial optimization problems. Authored by Giorgio Ausiello, Alberto Marchetti-Spaccamela, Pierluigi Crescenzi, and Viggo Kann, the book is structured into several chapters that cover fundamental concepts, design techniques, and advanced topics in approximation algorithms. The first chapter introduces the complexity of optimization problems, including the analysis of algorithms, complexity classes, and reducibility among problems. It also discusses the complexity of optimization problems and their classification into PO (Polynomial Optimization), NPO (Non-Deterministic Polynomial Optimization), and NP-hard optimization problems. Subsequent chapters delve into various design techniques for approximation algorithms, such as the greedy method, sequential algorithms for partitioning problems, local search, linear programming-based algorithms, dynamic programming, and randomized algorithms. The book also explores the concept of approximation classes, including APX (Approximable Problems), PTAS ( Polynomial-Time Approximation Schemes), and FPTAS (Fully Polynomial-Time Approximation Schemes). The sixth to eighth chapters focus on theoretical aspects, including the PCP (Probabilistically Checkable Proofs) model, the PCP theorem, and the study of approximability classes. The ninth and tenth chapters extend the discussion to probabilistic analysis of approximation algorithms and heuristic methods, respectively. The book includes a rich compendium of over 200 problems, categorized into twelve subjects, with each entry providing the best and worst approximation results. It also features exercises and an extensive bibliography, making it suitable for both undergraduate and graduate students in computer science and mathematics. The authors acknowledge the contributions of numerous students and friends who helped refine the content and express gratitude to their colleague and friend, Marco Protasi, who passed away before the book's completion.