1. NP-Completeness (36.4-5) P: yes and no in pt NP: yes in pt NPH  NPC
HW: 36-2, p.962;
37.2-3, p.974
NP-hard
NPC P NP

2. Satisfiability Boolean formulas: x, (x  y) (x  y) (xy) (xy)
Satisfiability Problem (SP):
given a Boolean formula
is there any 0-1 input (0-1 assignment to variables) s.t. formula is true (=1)?
Cook’s Theorem: SP is NP-complete.
SP is an NP-problem (why?)
SP is NP-hard (w/o proof)

3. (l11  l12 ... l1s1) ...  (ln1  ln2 ... lnsn)
3-CNF
Conjunctive normal form (CNF) =
(l11  l12 ... l1s1) ...  (ln1  ln2 ... lnsn)
each literal l is either variable or negation
3-CNF: each si=3
3-CNF Satisfiability is NPC
(x1 x2)  (x1   x4 )
Corresponding tree
y3(y1  (x1  x2))
(y2  ( x1   x4 ))
Truth assignment for each clause using tables.
y3
y1
y2
 x1 x2 x1
 x4

4. Independent Set Independent set in a graph G:
pairwise nonadjacent vertices
Max Independent Set is NPC
Construct a graph G
literal = vertex
two vertices are adjacent iff
they are in the same clause
they are negations of each other
3-CNF with k clauses is satisfiable
iff G has independent k-set
assign 1’s to literals-vertices of independent set

5. MAX Clique and Vertex Cover
The maximum number of pairwise adjacent vertices
Vertex Cover:
the set of vertices which has at least
one endpoint in each edge
If C is vertex cover, then V - C is an independent set
MIN Vertex Cover is NPC