The Nature of Computation

Transcription

1 The Nature of Computation Cristopher Moore University of New Mexico, Albuquerque and Santa Fe Institute Stephan Mertens Otto-von-Guericke University, Magdeburg and Santa Fe Institute OXFORD UNIVERSITY PRESS

2 Contents Figure Credits Preface xv 1 Prologue Crossing Bridges Intractable Itineraries Playing Chess With God What Lies Ahead 10 Problems 11 Notes 13 2 The Basics 2.1 Problems and Solutions 2.2 Time, Space, and Scaling 2.3 Intrinsic Complexity 2.4 The Importance of Being Polynomial 2.5 Tractability and Mathematical Insight Problems Notes 3 Insights and Algorithms 3.1 Recursion 3.2 Divide and Conquer 3.3 Dynamic Programming 3.4 Getting There From Here 3.5 When Greed is Good 3.6 Finding a Better Flow 3.7 Flows, Cuts, and Duality 3.8 Transformations and Reductions Problems Notes vii

3 viii CONTENTS 4 Needles in a Haystack: the Class NP Needles and Haystacks A Tour of NP Search, Existence, and Nondeterminism Knots and Primes 115 Problems 121 Notes Who is the Hardest One of All? NP-Completeness 5.1 When One Problem Captures Them All 5.2 Circuits and Formulas 5.3 D esigning Reductions 5.4 Completeness as a Surprise 5.5 The Boundary Between Easy and Hard 5.6 Finally, Hamiltonian Path Problems Notes The Deep Question: P vs. NP What if P -=NP? Upper Bounds are Easy, Lower Bounds Are Hard Diagonalization and the Time Hierarchy Possible Worlds Natural Proofs Problems in the Gap Nonconstructive Proofs The Road Ahead 210 Problems 211 Notes The Grand Unified Theory of Computation Babbage's Vision and Hilbert's Dream Universality and Undecidability Building Blocks: Recursive Functions Form is Function: the 2.-Calculus Turing's Applied Philosophy Computation Everywhere 264 Problems 284 Notes Memory, Paths, and Garnes Welcome to the State Space Show Me The Way L and NL-Completeness 310

4 CONTENTS ix 8.4 Middle-First Search and Nondeterministic Space You Can't Get There From Here PSPACE, Garnes, and Quantified SAT Garnes People Play Symmetric Space 339 Problems 341 Notes Optimization and Approximation Three Flavors of Optimization Approximations Inapproximability Jewels and Facets: Linear Programming Through the Looking-Glass: Duality Solving by Balloon: Interior Point Methods Hunting with Eggshells Algorithmic Cubism Trees, Tours, and Polytopes Solving Hard Problems in Practice 414 Problems 427 Notes Randomized Algorithms Foiling the Adversary The Smallest Cut The Satisfied Drunkard: Wa1kSAT Solving in Heaven, Projecting to Earth Garnes Against the Adversary Fingerprints, Hash Functions, and Uniqueness The Roots of Identity Primality Randomized Complexity Classes 488 Problems 491 Notes Interaction and Pseudorandomness The Tale of Arthur and Merlin The Fable of the Chess Master Probabilistically Checkable Proofs Pseudorandom Generators and Derandomization 540 Problems 553 Notes 560

5 x CONTENTS 12 Random Walks and Rapid Mixing A Random Walk in Physics The Approach to Equilibrium Equilibrium Indicators Coupling Coloring a Graph, Randomly Burying Ancient History: Coupling from the Past The Spectral Gap Flows of Probability: Conductance Expanders Mixing in Time and Space 623 Problems 626 Notes Counting, Sampling, and Statistical Physics Spanning Trees and the Determinant Perfect Matchings and the Permanent The Complexity of Counting From Counting to Sampling, and Back Random Matchings and Approximating the Permanent Planar Graphs and Asymptotics an Lattices Solving the Ising Model 693 Problems 703 Notes When Formulas Freeze: Phase Transitions in Computation Experiments and Conjectures Random Graphs, Giant Components, and Cores Equations of Motion: Algorithmic Lower Bounds Magic Moments The Easiest Hard Problem Message Passing Survey Propagation and the Geometry of Solutions Frozen Variables and Hardness 793 Problems 796 Notes Quantum Computation Particles, Waves, and Amplitudes States and Operators Spooky Action at a Distance Algorithmic Interference Cryptography and Shor's Algorithm Graph Isomorphism and the Hidden Subgroup Problem 862

6 CONTENTS )d 15.7 Quantum Haystacks: Grover's Algorithm Quantum Walks and Scattering 876 Problems 888 Notes 902 A Mathematical Tools 911 A.1 The Story of A.2 Approximations and Inequalities 914 A.3 Chance and Necessity 917 A.4 Dice and Drunkards 923 A.5 Concentration Inequalities 927 A.6 Asymptotic Integrals 931 A.7 Groups, Rings, and Fields 933 Problems 939 References 945 Index 974