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co-NP and the Asymmetry of NP

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co-NP and the Asymmetry of NP课程教案、知识点、字幕

与NP类相对应

还有一个重要的复杂性分类

co-NP类

我们做一个简要的介绍

在NP类的定义中

我们要求对于是实例有一个短的证明

我们来看几个例子

适定性问题和恒真命题

SAT问题可以通过验证一组真值分配

来证明CNF是适定的

恒真命题是要证明一个CNF

对于任意赋值都是正确的 或错误的

如何来验证它呢

对于哈密尔顿圈问题

我们通过检验所给的圈是哈密尔顿圈

来验证它的实例是是实例

但如何去验证一个图不存在哈密尔顿圈呢

我们知道 SAT是NP完全问题

但如何来刻画恒真命题的分类呢

我们定义NP为可以多项式时间检验的

判定问题的集合

包括SAT 哈密尔顿圈问题

合数问题等等

给定一个判定问题X

我们定义它的补问题X bar

为同一个问题

但是有相反的回答

如X={2, 3, 5, 7, 11, 13, 17, 23, 29, …}

为素数的集合

它的补={0, 1, 4, 6, 8, 9, 10, 12, 14, 15, …}

我们定义co-NP为

NP问题的补问题构成的集合

恒真命题 无哈密尔顿圈问题 素数问题

都属于co-NP

一个基础性的问题NP=co-NP吗

也就是说是实例有短证明

当且仅当否实例也有短证明吗

普遍认为它们是不等的

对于NP和co-NP

我们有下面的结论

如果NP≠co-NP

那么P≠NP

P问题的补也属于P

如果P=NP的话

那么NP问题的补问题

即co-NP问题也属于P

从而NP=co-NP=P

与已知矛盾

算法设计与分析课程列表:

1 Introduction of Algorithm

-1.1 Introduction

--Introduction

-1.2 A First Problem: Stable Matching

--A First Problem: Stable Matching

-1.3 Gale-Shapley Algorithm

--Gale-Shapley Algorithm

-1.4 Understanding Gale-Shapley Algorithm

--Understanding Gale-Shapley Algorithm

-Homework1

-Lecture note 1

--Lecture note 1 Introduction of Algorithm

2 Basics of Algorithm Analysis

-2.1 Computational Tractability

--Computational Tractability

-2.2 Asymptotic Order of Growth

--Asymptotic Order of Growth

-2.3 A Survey of Common Running Times

--A Survey of Common Running Times

-Homework2

-Lecture note 2

--Lecture note 2 Basics of Algorithm Analysis

3 Graph

-3.1 Basic Definitions and Applications

--Basic Definitions and Applications

-3.2 Graph Traversal

--Graph Traversal

-3.3 Testing Bipartiteness

--Testing Bipartiteness

-3.4 Connectivity in Directed Graphs

--Connectivity in Directed Graphs

-3.5 DAG and Topological Ordering

--DAG and Topological Ordering

-Homework3

-Lecture note 3

--Lecture note 3 Graph

4 Greedy Algorithms

-4.1 Coin Changing

--Coin Changing

-4.2 Interval Scheduling

--Interval Scheduling

-4.3 Interval Partitioning

--Interval Partitioning

-4.4 Scheduling to Minimize Lateness

--Scheduling to Minimize Lateness

-4.5 Optimal Caching

--Optimal Caching

-4.6 Shortest Paths in a Graph

--Shortest Paths in a Graph

-4.7 Minimum Spanning Tree

--Minimum Spanning Tree

-4.8 Correctness of Algorithms

--Correctness of Algorithms

-4.9 Clustering

--Clustering

-Homework4

-Lecture note 4

--Lecture note 4 Greedy Algorithms

5 Divide and Conquer

-5.1 Mergesort

--Mergesort

-5.2 Counting Inversions

--Counting Inversions

-5.3 Closest Pair of Points

--Closest Pair of Points

-5.4 Integer Multiplication

--Integer Multiplication

-5.5 Matrix Multiplication

--Video

-5.6 Convolution and FFT

--Convolution and FFT

-5.7 FFT

--FFT

-5.8 Inverse DFT

--Inverse DFT

-Homework5

-Lecture note 5

--Lecture note 5 Divide and Conquer

6 Dynamic Programming

-6.1 Weighted Interval Scheduling

--Weighted Interval Scheduling

-6.2 Segmented Least Squares

--Segmented Least Squares

-6.3 Knapsack Problem

--Knapsack Problem

-6.4 RNA Secondary Structure

--RNA Secondary Structure

-6.5 Sequence Alignment

--Sequence Alignment

-6.6 Shortest Paths

--Shortest Paths

-Homework6

-Lecture note 6

--Lecture note 6 Dynamic Programming

7 Network Flow

-7.1 Flows and Cuts

--Flows and Cuts

-7.2 Minimum Cut and Maximum Flow

--Minimum Cut and Maximum Flow

-7.3 Ford-Fulkerson Algorithm

--Ford-Fulkerson Algorithm

-7.4 Choosing Good Augmenting Paths

--Choosing Good Augmenting Paths

-7.5 Bipartite Matching

--Bipartite Matching

-Homework7

-Lecture note 7

--Lecture note 7 Network Flow

8 NP and Computational Intractability

-8.1 Polynomial-Time Reductions

--Polynomial-Time Reductions

-8.2 Basic Reduction Strategies I

--Basic Reduction Strategies I

-8.3 Basic Reduction Strategies II

--Basic Reduction Strategies II

-8.4 Definition of NP

--Definition of NP

-8.5 Problems in NP

--Problems in NP

-8.6 NP-Completeness

--NP-Completeness

-8.7 Sequencing Problems

--Sequencing Problems

-8.8 Numerical Problems

--Numerical Problems

-8.9 co-NP and the Asymmetry of NP

--co-NP and the Asymmetry of NP

-Homework8

-Lecture note 8

--Lecture note 8 NP and Computational Intractability

9 Approximation Algorithms

-9.1 Load Balancing

--Load Balancing

-9.2 Center Selection

--Center Selection

-9.3 The Pricing Method: Vertex Cover

--The Pricing Method: Vertex Cover

-9.4 LP Rounding: Vertex Cover

--LP Rounding: Vertex Cover

-9.5 Knapsack Problem

--Knapsack Problem

-Homework9

-Lecture note 9

--Lecture note 9 Approximation Algorithms

10 Local Search

-10.1 Landscape of an Optimization Problem

--Landscape of an Optimization Problem

-10.2 Maximum Cut

--Maximum Cut

-10.3 Nash Equilibria

--Nash Equilibria

-10.4 Price of Stability

--Price of Stability

-Homework10

-Lecture note 10

--Lecture note 10 Local Search

11 Randomized Algorithms

-11.1 Contention Resolution

--Contention Resolution

-11.2 Linearity of Expectation

--Linearity of Expectation

-11.3 MAX 3-SAT

--MAX 3-SAT

-11.4 Chernoff Bounds

--Chernoff Bounds

-Homework11

-Lecture note 11

--Lecture note 11 Randomized Algorithms

Exam

-Exam

co-NP and the Asymmetry of NP笔记与讨论

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