P VS NP

Posted 2022-01-09 mb61caa1c74a413

目录

• 1.1 P
  • 1.1.1 NP
  • 1.1.2 Certifiers and certificates: satisfiability
• 2.1 Certifiers and certificates: Hamilton path
  • 2.1.1 Some problems in NP
• 3.1 P, NP, and EXP
  • 3.1.1 The main question: P vs. NP

1.1 P

Def. Algorithm A runs in polynomial time if for every string s, A(s)
terminates in ≤ p( ⎢s⎢) “steps,” where p(⋅) is some polynomial function.

1.1.1 NP

Def. Algorithm C(s, t) is a certifier for problem X if for every string s : s ∈ X iff there exists a string t such that C(s, t) = yes.
NP = set of decision problems for which there exists a poly-time certifier.
・C(s, t) is a poly-time algorithm.
・Certificate t is of polynomial size: ⎢t⎢ ≤ p(⎢s⎢) for some polynomial p(⋅)

关于NP举一个例子：

• 我们找是否有两个数相乘等于437669
• 首先我们找到了一个可验证的解就是541
• 那么只要我们可以找多项式时间内找到809我们就可以找到原问题的解。

1.1.2 Certifiers and certificates: satisfiability

• 3-SAT. SAT where each clause contains exactly 3 literals.
• Certificate. An assignment of truth values to the Boolean variables.
• Certifier. Check that each clause in Φ has at least one true literal.

验证步骤是这样的：

1. 首先我们找到其中一个解作为验证实例比如x1=true,x2=tru2,x3=false,x4=false;
2. 我们必须在多项式时间内可以验证这个解是成立的！

2.1 Certifiers and certificates: Hamilton path

HAMILTON-PATH： Given an undirected graph G = (V, E), does there exist a
simple path P that visits every node?

Certificate： A permutation π of the n nodes.

Certifier： Check that π contains each node in V exactly once,
and that G contains an edge between each pair of adjacent nodes

2.1.1 Some problems in NP

3.1 P, NP, and EXP

P. Decision problems for which there exists a poly-time algorithm.
NP. Decision problems for which there exists a poly-time certifier.
EXP. Decision problems for which there exists an exponential-time algorithm.

3.1.1 The main question: P vs. NP

Does P = NP? [Cook 1971, Edmonds, Levin, Yablonski, Gödel]
Is the decision problem as easy as the certification problem?

If yes… Efficient algorithms for 3-SAT, TSP, VERTEX-COVER, FACTOR, …
If no… No efficient algorithms possible for 3-SAT, TSP, VERTEX-COVER, …