当前课程知识点:算法设计与分析 > 8 NP and Computational Intractability > 8.9 co-NP and the Asymmetry of NP > co-NP and the Asymmetry of NP
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
-1.1 Introduction
-1.2 A First Problem: Stable Matching
--A First Problem: Stable Matching
-1.3 Gale-Shapley Algorithm
-1.4 Understanding Gale-Shapley Algorithm
--Understanding Gale-Shapley Algorithm
-Homework1
-Lecture note 1
--Lecture note 1 Introduction of Algorithm
-2.1 Computational Tractability
-2.2 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
-3.1 Basic Definitions and Applications
--Basic Definitions and Applications
-3.2 Graph Traversal
-3.3 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
-4.1 Coin Changing
-4.2 Interval Scheduling
-4.3 Interval Partitioning
-4.4 Scheduling to Minimize Lateness
--Scheduling to Minimize Lateness
-4.5 Optimal Caching
-4.6 Shortest Paths in a Graph
-4.7 Minimum Spanning Tree
-4.8 Correctness of Algorithms
-4.9 Clustering
-Homework4
-Lecture note 4
--Lecture note 4 Greedy Algorithms
-5.1 Mergesort
-5.2 Counting Inversions
-5.3 Closest Pair of Points
-5.4 Integer Multiplication
-5.5 Matrix Multiplication
--Video
-5.6 Convolution and FFT
-5.7 FFT
--FFT
-5.8 Inverse DFT
-Homework5
-Lecture note 5
--Lecture note 5 Divide and Conquer
-6.1 Weighted Interval Scheduling
--Weighted Interval Scheduling
-6.2 Segmented Least Squares
-6.3 Knapsack Problem
-6.4 RNA Secondary Structure
-6.5 Sequence Alignment
-6.6 Shortest Paths
-Homework6
-Lecture note 6
--Lecture note 6 Dynamic Programming
-7.1 Flows and Cuts
-7.2 Minimum Cut and Maximum Flow
--Minimum Cut and Maximum Flow
-7.3 Ford-Fulkerson Algorithm
-7.4 Choosing Good Augmenting Paths
--Choosing Good Augmenting Paths
-7.5 Bipartite Matching
-Homework7
-Lecture note 7
-8.1 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
-8.5 Problems in NP
-8.6 NP-Completeness
-8.7 Sequencing Problems
-8.8 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
-9.1 Load Balancing
-9.2 Center Selection
-9.3 The Pricing Method: Vertex Cover
--The Pricing Method: Vertex Cover
-9.4 LP Rounding: Vertex Cover
-9.5 Knapsack Problem
-Homework9
-Lecture note 9
--Lecture note 9 Approximation Algorithms
-10.1 Landscape of an Optimization Problem
--Landscape of an Optimization Problem
-10.2 Maximum Cut
-10.3 Nash Equilibria
-10.4 Price of Stability
-Homework10
-Lecture note 10
--Lecture note 10 Local Search
-11.1 Contention Resolution
-11.2 Linearity of Expectation
-11.3 MAX 3-SAT
-11.4 Chernoff Bounds
-Homework11
-Lecture note 11
--Lecture note 11 Randomized Algorithms
-Exam