Satisfiability

Posted 2022-01-09 mb61caa1c74a413

目录

• 1.1 SAT
  • 1.1.1 3-SAT
  • 1.1.2 3-satisfiability reduces to independent set
  • 1.1.3 必要性
  • 1.1.4 充分性

1.1 SAT

SAT. Given a CNF formula Φ, does it have a satisfying truth assignment?
可满足性要求的是结果为TRUE！

1.1.1 3-SAT

SAT： where each clause contains exactly 3 literals
(and each literal corresponds to a different variable).

先来看一个列子：

当然一个正确的实例是非常好找到的！（那么有没有可能永远找不到解？）答案是肯定！原因见下文。

1.1.2 3-satisfiability reduces to independent set

Pf. Given an instance Φ of 3-SAT, we construct an instance (G, k) of
INDEPENDENT-SET that has an independent set of size k = ⎜Φ⎜ iff Φ is satisfiable.

定理：3元可满足性问题可以规约到独立集问题。

Construction.

• G contains 3 nodes for each clause, one for each literal.
• Connect 3 literals in a clause in a triangle.
• Connect literal to each of its negations

1.1.3 必要性

定理：Φ is satisfiable iff G contains an independent set of size k = ⎜Φ⎜.
Pf.
⇐ Consider any satisfying assignment for Φ.

• Select one true literal from each clause/triangle.
• This is an independent set of size k = ⎜Φ⎜.
  

1.1.4 充分性

定理：Φ is satisfiable iff G contains an independent set of size k = ⎜Φ⎜.
Pf. ⇒ Let S be independent set of size k.

• S must contain exactly one node in each triangle.
• Set these literals to true (and remaining literals consistently).
• All clauses in Φ are satisfied.