Problems in NP-complete, Complexity P & NP

Hundreds of well-known problems are NP-complete:

3-CNF SAT is the beginning point of chains of trouble reduction theorems to exhibit that hundreds of other well known decision troubles are NP complete. The problem CLIQUE is an illustration: Given a graph G and an integer k, does G comprise a clique of size ≥ k?

Theorem: CLIQUE is NP-complete

Proof: Show that the 3-CNF ≤p CLIQUE. Given 3-CNF expression F, build a graph G = (V, E) and an integer k in such a way that F is satisfiable if and only if G consists of a k-clique.

Suppose F = (z11 ∨ z12 ∨ z13) ∧ (z21 ∨ z22 ∨ z23) ∧ ... ∧ (zm1 ∨ zm2 ∨ zm3), where each zij is the literal.

To each of the occurrence of literal we assign a vertex, that is, V = {(1,1), (1,2), (1,3), ... , (m, 1), (m, 2), (m, 3)}

We introduce an edge ((i, j) (p, q)) if and only if i ≠ p (that is, the two literals are in distinct clauses) and zij ≠ ¬zpq (that is, the 2 literals don’t clash, that is, both can be made true beneath similar assignment).

Lastly, suppose k, the desired clique size, be = m, the number of clauses.

With this construction of G we examine that F is satisfiable through an assignment A

If and only if 1) All clause includes a literal which is true beneath A, state z1, j1, z2, j2, ... ,  zm, jm
If and only if 2) there are literals z1, j1,  z2, j2, ... , zm, jm no two of which are negations of one other
If and only if 3) there are vertices (1, j1), (2, j2), ... , (m, jm) which are pair-wise joined by an edge
If and only if 4) G consists of a k-clique.

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