Lecture 10: NP¶
1 Easy vs. Hard¶
The easist: \(O(N)\), since we have to read all the input.
The hardest: Undecidable problems, such as the Halting Problem.
2 The Class NP¶
Turing machine is introduced to simulate any kind of computation which a mathematician can do by some arithmetical method. 图灵机由无限长的纸带和读写头组成，纸带上的每个格子都可以存储一个符号，读写头可以读取或写入符号。
A Deterministic Turing Machine executes one instruction at each point in time. Then depending on the the instruction, it goes to the next unique instruction.
A Non-deterministic Turing Machine is free to choose its next step from a inite set. And if one of these steps leads to a solution, it will always choose the correct one.
这里的 NP 是 Non-deterministic Polynomial time 的缩写。
The problem is NP if we can prove any solution is true in polynomial time.
注意
Not all decidable problems are in NP. For example, consider the problem of determing whether a graph does not contain a Hamiltonian cycle.
按照以上的叙述，可以知道 \(P \subseteq NP\)。但是有没有 \(P \subset NP\) 呢？这仍然是一个未解决的问题。
3 NP-Complete Problems: the Hardest in NP¶
An NP-Complete problem has the property that any problem in NP can be polynomially reduced to it.
If we can solve one NP-Complete problem in polynomial time, we can solve all NP problems in polynomial time.
Given any instance \(\alpha\) of an NP-Complete problem, we can reduce it to an instance \(\beta\) of another NP-Complete problem in polynomial time, and then solve \(\beta\) in polynomial time, then we can solve \(\alpha\) in polynomial time.
提示
Decision problem is easier, since we only have to answer yes or no. Optimization problem can be related to a decision problem. Example: SHORTEST-PATH relates to PATH: given a directed graph G, vertices u and v, and an integer k, does a path exist from u to v consisting of at most k edges?
第一个被证明是 NP-Complete 的问题是 SAT（Satisfiability）问题。
4 A Formal-language Framework¶
An abstract problem Q is a binary relation on a set I of problem instances and a set S of solutions.
If we map I into a binary string {0,1}, then Q is a concrete problem*.
对于 decision problem，如下图：
有关 co-NP 的定义：
