Introduction to Probabilistically Checkable Proofs and Hardness of Approximation

Theoretical computer science contains some of the most profound mathematical discoveries of the modern era, yet advanced proof systems can feel out of reach. This course offers a clear, structured path to understanding Probabilistically Checkable Proofs (PCPs) and their revolutionary impact on computational complexity. You will transition from basic complexity theory to grasping how local verification of mathematical proofs reshapes our understanding of optimization problems. What you'll learn: Understand the foundational definitions of NP, PCP classes, and the core statement of the PCP Theorem; Explore the algebraic proof techniques used to construct basic probabilistically checkable proofs; Analyze the direct connection between proof verification and the hardness of approximation algorithms; Study the role of modern concepts like the Unique-Games Conjecture and hardness amplification; Examine the mechanics of error-correcting codes and the long-code framework in proof systems. The course begins with essential terminology, complexity classes, and foundational definitions before guiding you through algebraic constructions and the mathematical proofs that define modern complexity theory. Designed for beginners in theoretical computer science, this text-only program requires no advanced prerequisites beyond basic discrete mathematics and linear algebra. Start reading today to unlock the deepest principles of modern computational complexity.