1. NP-Completeness (36.4-5/34.4-5)
P: yes and no in pt
NP: yes in pt
NPH  NPC
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 x z y’ x’ z’ a a’ x y x, z’, y independent
Independent set in a graph G:
pairwise nonadjacent vertices
Max Independent Set is NPC
Is there independent set of size k?
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
Example: f = (x+z+y’) & (x’+z’+a) & (a’+x+y) x z y’ x’ z’ a a’ x y
x, z’, y independent
F is satisfiable:
f = 1 if x = z’ = y = 1

5. MAX Clique G Complement G’ Max Clique (MC): MC is in NP MC is in NPC
Find the maximum number of pairwise adjacent vertices
MC is in NP
for the answer yes there is certificate of polynomial length = clique which can be checked in polynomial time
MC is in NPC
Polynomial time reduction from MIS:
For any graph G any independent set in G 1-1 corresponds to clique in the complement graph G’
red independent set
red clique G
noedge  edge
edge  noedge
Complement G’

6. Minimum Vertex Cover Vertex Cover: Minimum Vertex Cover (MVC):
the set of vertices which has at least one endpoint in each edge
Minimum Vertex Cover (MVC):
MIN Vertex Cover is NPC
If C is vertex cover, then V - C is an independent set
red independent set
blue vertex cover

7. Set Cover d a b c A B A = {a,b,c}, B = {c,d}
Given: a set X and a family F of subsets of X, F  2X, s.t. X covered by F
Find : subfamily G of F such that G covers X and |G| is minimize
Set Cover is NPC
reduction from Vertex Covert
Graph representation:
edge between set A F and element x  X means x  A d a b c
red elements of ground set X
blue subsets in family F A B
A = {a,b,c}, B = {c,d}

8. Intermediate Classes NP-hard NPC Dense Set Cover is NP but not in P
neither in NPC
Dense Set Cover:
Each element of X belongs to at least half of all sets in F

9. Runtime Complexity Classes
Runtime order:
constant
almost constant
logarithmic
sublinear
linear
pseudolinear
quadratic
polynomial
subexponential
exponential
superexponential
Example
adding an element in a queue/stack
inverse Ackerman function = O(loglog…log n)
n times
extracting minimum from binary heap
n1/2
traversing binary search tree, list
O(n log n) sorting n numbers, closest pair, MST, Dijkstra shortest paths
adding two nn matrices
e n ^ (1/2)
e n, , n!
Ackerman function 2 .
n times