ISBN: 3-540-64201-3
TITLE: Lectures on Proof Verification and Approximation Algorithms
AUTHOR: Mayr, Ernst W.; Prmel, Hans Jrgen; Steger, Angelika (Eds.)
TOC:

Introduction 1 
1. Introduction to the Theory of Complexity and Approximation 
Algorithms 5 
Thomas Jansen 
1.1 Introduction 5 
1.2 Basic Definitions 6 
1.3 Complexity Classes for Randomized Algorithms 12 
1.4 Optimization and Approximation 16 
2. Introduction to Randomized Algorithms 29 
Artur Andrzejak 
2.1 Introduction 29 
2.2 Las Vegas and Monte Carlo Algorithms 30 
2.3 Examples of Randomized Algorithms 32 
2.4 Randomized Rounding of Linear Programs 34 
2.5 A Randomized Algorithm for MaxSat with Expected 
Performance Ratio 4/3 37 
3. Derandomization 41 
Detlef Sieling 
3.1 Introduction 41 
3.2 The Method of Conditional Probabilities 42 
3.3 Approximation Algorithms for MaxEkSat, MaxSat and 
MaxLinEq3-2 44 
3.4 Approximation Algorithms for Linear Integer Programs 46 
3.5 Reduction of the Size of the Probability Space 53 
3.6 Another Approximation Algorithm for MaxE3Sat 55 
3.7 A PRAM Algorithm for MaximalIndependentSet 56 
4. Proof Checking and Non-approximability 63 
Stefan Hougardy 
4.1 Introduction 63 
4.2 Probabilistically Checkable Proofs 63 
4.3 PCP and Non-approximability 66 
4.4 Non-approximability of APX-Complete Problems 69 
4.5 Expanders and the Hardness of Approximating MaxE3Sat-b 71 
4.6 Non-approximability of MaxClique 73 
4.7 Improved Non-approximability Results for MaxClique 77 
5. Proving the PCP-Theorem 83 
Volker Heun, Wolfgang Merkle, Ulrich Weigand 
5.1 Introduction and Overview 83 
5.2 Extended and Constant Verifiers 85 
5.3 Encodings 88 
5.4 Efficient Solution Verifiers 109 
5.5 Composing Verifiers and the PCP-Theorem 121 
5.6 The LFKN Test 134 
5.7 The Low Degree Test 141 
5.8 A Proof of Cook's Theorem 157 
6. Parallel Repetition of MIP(2,1) Systems 161 
Clemens Grpl, Martin Skutella 
6.1 Prologue 161 
6.2 Introduction 161 
6.3 Two-Prover One-Round Proof Systems 162 
6.4 Reducing the Error Probability 164 
6.5 Coordinate Games 166 
6.6 How to Prove the Parallel Repetition Theorem (I) 168 
6.7 How to Prove the Parallel Repetition Theorem (II) 171 
7. Bounds for Approximating MAXLINEQ3-2 and MAXEkSAT 179 
Sebastian Seibert, Thomas Wilke 
7.1 Introduction 179 
7.2 Overall Structure of the Proofs 180 
7.3 Long Code, Basic Tests, and Fourier Transform 181 
7.4 Using the Parallel Repetition Theorem for Showing Satisfiability 189 
7.5 An Optimal Lower Bound for Approximating MaxLinEq3-2 192 
7.6 Optimal Lower Bounds for Approximating MaxEkSat 198 
8. Deriving Non-approximability Results by Reductions 213 
Claus Rick, Hein Rhrig 
8.1 Introduction 213 
8.2 Constraint Satisfaction Problems 215 
8.3 The Quality of Gadgets 217 
8.4 Weighted vs. Unweighted Problems 221 
8.5 Gadget Construction 225 
8.6 Improved Results 230 
9. Optimal Non-approximability of MAXCLIQUE 235 
Martin Mundhenk, Anna Slobodov 
9.1 Introduction 235 
9.2 A PCP-System for RobE3Sat and Its Parallel Repetition 236 
9.3 The Long Code and Its Complete Test 239 
9.4 The Non-approximability of MaxClique 243 
10. The Hardness of Approximating Set Cover 249 
Alexander Wolff 
10.1 Introduction 249 
10.2 The Multi-prover Proof System 252 
10.3 Construction of a Partition System 255 
10.4 Reduction to Set Cover 256 
11. Semidefinite Programming and Its Applications to 
Approximation Algorithms 263 
Thomas Hofmeister, Martin Hhne 
11.1 Introduction 263 
11.2 Basics from Matrix Theory 267 
11.3 Semidefinite Programming 270 
11.4 Duality and an Interior-Point Method 274 
11.5 Approximation Algorithms for MaxCut 278 
11.6 Modeling Asymmetric Problems 285 
11.7 Combining Semidefinite Programming with 
Classical Approximation Algorithms 289 
11.8 Improving the Approximation Ratio 293 
11.9 Modeling Maximum Independent Set as a Semidefinite Program? 296 
12. Dense Instances of Hard Optimization Problems 299 
Katja Wolf 
12.1 Introduction 299 
12.2 Motivation and Preliminaries 300 
12.3 Approximating Smooth Integer Programs 304 
12.4 Polynomial Time Approximation Schemes for Dense MaxCut 
and MaxEkSat Problems 308 
12.5 Related Work 310 
13. Polynomial Time Approximation Schemes for Geometric 
Optimization Problems in Euclidean Metric Spaces 313 
Richard Mayr, Annette Schelten 
13.1 Introduction 313 
13.2 Definitions and Notations 314 
13.3 The Structure Theorem and Its Proof 316 
13.4 The Algorithm 320 
13.5 Applications to Some Related Problems 321 
13.6 The Earlier PTAS 322 
Bibliography 325 
Author Index 335 
Subject Index 337 
List of Contributors 343 
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