## The Nature of ComputationComputational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again. |

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### Contents

Figure Credits | |

The Basics | |

Insights and Algorithms | |

the ClassNP | |

Who is the HardestOne ofAll? NPCompleteness | |

P vs | |

8 | |

Optimization andApproximation 9 1 Three Flavors of Optimization 9 2 Approximations | |

Interaction | |

Random Walks andRapidMixing 12 1 A Random Walk inPhysics 12 2 The Approach | |

Counting Sampling andStatistical Physics 13 1 Spanning Trees | |

PhaseTransitions inComputation 14 1 Experiments and Conjectures 14 2 Random GraphsGiant Components and Cores | |

Quantum Computation | |

A Mathematical Tools A 1 The Storyof O A 2 Approximations and Inequalities A 3 Chance and Necessity A 4 Dice and Drunkards A 5 Concentratio... | |

Randomized Algorithms | |

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### Common terms and phrases

3COLORING approximation Arthur bits choose circuit clauses color complexity computation constant constraints corresponding cycle define denote deterministic edges eigenvalues equations expected number exponentially factor factor graph finite flip formula Fourier gadget given gives graph G Halting Problem HAMILTONIAN PATH Hint independent set input instance integer INTEGER PARTITIONING inthe Ising model lattice LINEAR PROGRAMMING lower bound Markov chain mathematical matrix MAX CUT Merlin minimum spanning tree move neighbors NPcomplete ofthe onthe optimal pair perfect matchings permutation planar graph poly(n polynomial polynomialtime algorithm polytope possible prime probability distribution problem proof prove pseudorandom PSPACE quantum qubit random walk randomized algorithm REACHABILITY reduction running satisfying assignments Section shown in Figure simulate solution solve spanning tree step string subset Suppose Theorem tiles Turing machine variables vector VERTEX COVER vertices wecan weight yesinstance