Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)

These are the lecture notes for the DIMACS Tutorial “Limits of Approximation Algorithms: PCPs and Unique Games” held at the DIMACS Center, CoRE Building, Rutgers University on 20-21 July, 2009. This tutorial was jointly sponsored by the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus on Algorithmic Foundations of the Internet, and the Center for Computational Intractability with support from the National Security Agency and the National Science Foundation. The speakers at the tutorial were Matthew Andrews, Sanjeev Arora, Moses Charikar, Prahladh Harsha, Subhash Khot, Dana Moshkovitz and Lisa Zhang. The sribes were Ashkan Aazami, Dev Desai, Igor Gorodezky, Geetha Jagannathan, Alexander S. Kulikov, Darakhshan J. Mir, Alantha Newman, Aleksandar Nikolov, David Pritchard and Gwen Spencer.

💡 Research Summary

The document is a comprehensive set of lecture notes from the 2009 DIMACS tutorial “Limits of Approximation Algorithms: PCPs and Unique Games.” It is organized into several parts that together provide a panoramic view of both the algorithmic techniques that yield good approximations for NP‑hard problems and the complexity‑theoretic barriers that limit how well such problems can be approximated.

The first section introduces the basic notions of approximation algorithms, defining approximation ratios for minimization and maximization problems and presenting classic examples. It explains the 2‑approximation for metric TSP based on doubling a minimum spanning tree, and the improvement to a 1.5‑approximation using Christofides’ matching step. The notion of a polynomial‑time approximation scheme (PTAS) is then explored in depth. Two families of PTAS are distinguished: Type‑1 schemes that rely on rounding values and dynamic programming (illustrated by the knapsack problem) and Type‑2 schemes that simplify the structure of the solution (illustrated by Arora’s quadtree‑based PTAS for Euclidean TSP).

The second part turns to hardness of approximation, beginning with the PCP theorem. It defines gap problems, explains how locally checkable proofs give rise to hardness results, and introduces projection games as a bridge between PCPs and gap problems. The notes discuss robust PCPs, 2‑projection PCPs, and the equivalence between them. Concrete locally testable codes such as Reed‑Muller codes, low‑degree tests, and zero‑sub‑cube tests are presented, showing how a verifier can check a proof by querying only a few bits.

The third section surveys approximation algorithms for several network design problems: minimum‑cost Steiner forest, fractional and integral congestion minimization, edge‑disjoint paths, and minimum‑cost network design. A detailed hardness analysis for the edge‑disjoint paths (EDP) problem is provided, covering both directed and undirected variants, reductions from bounded‑degree independent set, and the construction of special graphs G and H that encode the difficulty.

Sections four and five constitute a two‑part proof of the PCP theorem. Part I develops a robust PCP for CIRCUIT‑SAT, covering arithmetization of assignments and circuits, the design of a verifier, and the composition of PCPs to amplify soundness. Part II presents Håstad’s celebrated 3‑bit PCP. It explains the long code, the consistency test, the construction of the final test, and how these components combine to give a PCP with only three queries and optimal hardness parameters.

Section six introduces semidefinite programming (SDP) and its connection to Unique Games. Unique Games are defined as constraint satisfaction problems where each constraint is a permutation on a small label set. The notes give examples (linear equations, MAX‑CUT), discuss the distinction between satisfiable and almost‑satisfiable instances, and present an SDP relaxation together with Trevisan’s rounding algorithm. Improvements to the approximation ratio and the consequences for hardness of approximation are discussed.

The final section, authored by Subhash Khot and collaborators, leverages the Unique Games Conjecture (UGC) to prove optimal hardness for MAX‑CUT. It reviews the Goemans‑Williamson SDP‑based algorithm (achieving a 0.878… approximation), introduces the “label cover” reduction that connects MAX‑CUT to Unique Games, and states the main result: assuming UGC, no polynomial‑time algorithm can beat the Goemans‑Williamson ratio. The proof relies on the “Majority is Stablest” theorem, a deep result from analysis of Boolean functions, and on a careful reduction that embeds the long code test into a MAX‑CUT instance.

Overall, the notes weave together algorithmic design (LP/SDP relaxations, rounding, dynamic programming, quadtree partitioning) and hardness techniques (PCP constructions, robust PCPs, composition, Unique Games) to give readers a solid grounding in why certain approximation ratios are achievable and why others are provably out of reach. The material is dense but self‑contained, making it a valuable reference for graduate students and researchers interested in the frontier of approximation algorithms and computational complexity.