Theorem 36.11: CLIQUE $\in$ NPC

1. CLIQUE $\in$ NP

To show CLIQUE in NP, we use set V' of vertices as a certificate.

Verifying is polynomial time, check whether for every pair u,v in V', the edge is in E ( pairs).

2. L' = 3-CNF-SAT

3. 3-CNF-SAT $\leq_P$ CLIQUE

Start with instance of 3-CNF-SAT (also called 3CNF).

Let f be 3CNF with k clauses, .

For r = 1,2,..,k, each clause has three distinct literals l₁^r, l₂^r, l₃^r.

Construct a graph G such that f is satisfiable iff G has a clique of size k.

For each C_r in f, put triple of vertices v₁^r, v₂^r, v₃^r in V.

Add edge (v_i^r, v_j^s) if

1.

v_i^r and v_j^s are in different triples (r $\neq$ s), and

2.

their corresponding literals are consistent (l_i^r is not the negation of l_j^s).