计算机问题求解笔记

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第四章

NP 的informal 定义

A decision problem is in NP if whenever the answer for a particular instance is “yes”,there is a simple proof of this fact.(simple proof means that we can check in polynomial time).

NP P ⊆

If a problem is in P ,it has a polynomial algorithm that is guaranteed to give the right answer.If the input is a yes-instance,the sequence of steps executed by this algorithm,which end with it returning “yes”,constitutes a proof of that fact.thus .

So,P is a set of decision problem solvable in polynomial time.

GRAPH k-COLORING

Input: a graph G

Question : is there a proper k-coloring of G?

Reduction

A is reducible to

B and write B A ≤,if there is a polynomial-time algorithm that translates

instances of A to instances of B.in that case,B is at least as hard as A.

Graph 3-coloring ≤planar graph 3-coloring

只有当k=3时，才可以这么做，当k>3时，graph k-coloring 不能规约到planar graph k-coloring. 对于平面图而言，增加染色数，图的可染色问题会变得简单。但是对于一般的图来说，增加染色数不会降低问题的难度。

Exercise ：prove that Graph k-coloring ≤graph(k+1)-coloring for any k .

Hint :add a vertex connected to all other k vertexes.

CNF:Conjunctive normal form

CNF is the AND of a series of clauses and each clause is the OR of a set of literals,where a literal is either a variable or its negation.

SAT(约束可满足问题)

Input: A CNF Boolean formula Question: is ϕ satisfiable?

Graph 3-coloring ≤ SAT

K-SAT

Input : A CNF Boolean formula ϕ ,where each clause contains k variables

Question:is ϕ satisfiable?

Integer Partitioning

Input : A list of positive integers Question : is there a balanced partition,i.e,a subset s.t. Subset sum

Input : A set of positive integers and an integer t Question : Does there exist a subset such that Independent Set Formally,given a graph G = (V,E),we say that a subset is independent if no two vertices in S are adjacent.

Input: A graph G and an integer k

Question: Does G have an independent set of size k or more?

Vertex Cover

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A subset is a vertex cover if for every edge,at least one of e’s endpoint is in S

Input: A graph G and an integer k

Question: Does G have a vertex cover of size k or less?

Clique(最大完全子图)

Input: A graph G and an integer k

Question: Does G have a clique of size k or more?

三者可以相互规约，S为独立集，则所有边集中每条边至少有一个端点在G-S中；从而，最大的独立集对应最小的点覆盖G-S；clique的补图P,即，反之亦然，则clique中的最大完全子图对应于补图P中的最大独立集。

Formal definition of NP

NP is the class of problems A of the following form:

X is a yes-instance of A if and only if there exists a such that is a yes-instance of B where B is a decision problem in P regarding pairs and where .

EXP = TIME()

NTIME(f(n)) is the class of problems A of the following form:

is a yes-instance of A if and only if there exists a such that is a yes-instance of B.where B is a decision problem in TIME(f(n)) regarding pairs and where.For instance,NEXP = NTIME().

More general,we have:

Another definition of NP:

NP is the class of properties A of the form:

Where B is in P and where .

Co-NP的定义

No Hamiltonian path

Input: A graph G

Question: is it true that G has no hamiltonian path?

This is a very different problem.there is a simple proof if the answer is “no”,but it seems hard to prove that the answer is “yes” without an exhausive search.problems like this have a complexity class of their own,called CoNP,the class of problems in which,if the input is a no-instance,there is a simple proof of that fact.

Primality

Input: A n-bit integer

Question: Is prime

记表示和模运算相等。

Suppose is a prime,if is an integer greater than 1 and less than ,its order is the smallest such that.

Fermat’s little theorem states that if is prime,then for any . this does not mean that the order of is .however,it does imply that is a divisor of .

Lehmer’s theorem

is prime if and only if an exists with order exactly .such an is called a primitive root.

验证是否为素数，只需要找出一个,满足的order是.

要证明的order为，要验证.这个问题可以简化为

，因为：

假设的order为，

,因为为整数