Black-box (oracle)
Feed me a weighted graph G
and I will tell you the weight
of the max-weight matching
of G.
Black-box (oracle)
5
2
2
Feed me a weighted graph G
and I will tell you the weight
of the max-weight matching
of G.
Black-box (oracle)
5
2
2
Feed me a weighted graph G
and I will tell you the weight
of the max-weight matching
of G.
5
Black-box (oracle)
Feed me a weighted graph G
and I will tell you the weight
of the max-weight matching
of G.
here is a graph G, find the
max-weight matching
G
Black-box (oracle)
here is a graph G,
find the max-weight
matching
Feed me a weighted graph G
and I will tell you the weight
of the max-weight matching
of G.
pick a vertex uV(G)
for each edge {u,v}E(G)
wundefined
if oracle(G-u-v) + w(u,v) = oracle (G) then wv
if w is undefined then recurse on (G-u)
else print({u,w}); recurse on (G-u-v)
3-SAT
x = variable
x = negation of a variable
literals
clause = disjunction of literals
xyz
x z
3-SAT
INSTANCE:
collection C of clauses, each clause
has at most 3 literals
QUESTION:
does there exist an assignment of
true/false to the variables which
satisfies all the clauses in C
3-SAT
INSTANCE:
collection C of clauses, each clause
has at most 3 literals
QUESTION:
does there exist an assignment of
true/false to the variables which
satisfies all the clauses in C
x y z
x  y  z
x  y
x
Independent Set
subset S of vertices such that no
two vertices in S are connected
Independent Set
subset S of vertices such that no
two vertices in S are connected
Independent Set
OPTIMIZATION VERSION:
INSTANCE: graph G
SOLUTION: independent set S in G
MEASURE: maximize the size of S
DECISION VERSION:
INSTANCE: graph G, number K
QUESTION: does G have independent
set of size  K
Independent Set  3-SAT
“is easier than”
if we have a black-box for 3-SAT
then we can solve Independent Set
in polynomial time
Independent Set reduces to 3-SAT
Independent Set  3-SAT
if we have a black-box for 3-SAT then we can solve
Independent Set in polynomial time
Give me a 3-SAT
formula and I will
tell you if it is satisfiable
We would like to solve the
Independent Set problem
using the black box in polynomial time.
Independent Set  3-SAT
Give me a 3-SAT
formula and I will
tell you if it is satisfiable
Graph G, K

3-SAT formula F
efficient transformation (i.e., polynomial – time)
G has independent set of size  K  F is satisfiable
Independent Set  3-SAT
Give me a 3-SAT
formula and I will
tell you if it is satisfiable
Graph G, K
V = {1,...,n}
E = edges

3-SAT formula F
variables x1,....,xn
 xi  xj for ij E
+ we need to ensure that  K of
the xi are TRUE
3-SAT  Independent Set
Give me a graph G and
a number K and I will
tell you if G has independent
set of size  K
3-SAT formula F  graph G, number K
3-SAT  Independent Set
Give me a graph G and
a number K and I will
tell you if G has independent
set of size  K
3-SAT formula F  graph G, number K
x  y  z
w  y  z
y
y
z
x
w
z
3-SAT  Independent Set
3-SAT formula F  graph G, number K
x  y  z
w  y  z
y
y
z
x
w
z
1) efficiently computable
2) F satisfiable   IS of size  m
3)  IS of size  m  F satisfiable
3-SAT  Independent Set
Independent Set  3-SAT
if 3-SAT is in P then Independent Set is in P
if Independent Set is in P then 3-SAT is in P
3-SAT
Independent Set
Many more reductions
Clique
Subset-Sum
3-COL
Planar 3-COL
Hamiltonian path
3-SAT
Independent Set
P and NP
P = decision problems that can be
solved in polynomial time.
NP = decision problems for which
the YES answer can be certified
and this certificate can be
verified in polynomial time.
NP = decision problems for which
the YES answer can be certified
and this certificate can be
verified in polynomial time.
3-SAT
Independent Set
NOT-3-SAT ?
NP = decision problems for which
the YES answer can be certified
and this certificate can be
verified in polynomial time.
COOK’S THEOREM
Every problem A NP
A  3-SAT
NP = decision problems for which
the YES answer can be certified
and this certificate can be
verified in polynomial time.
B is NP-hard
if every problem A NP
AB
B is NP-complete
if B is NP-hard, and
B is in NP
NP-hard
NP-complete
P
NP
Some NP-complete problems
Clique
Subset-Sum
3-COL
Planar 3-COL
Hamiltonian path
3-SAT
Independent Set
Clique
subset S of vertices such that every
two vertices in S are connected
Clique
INSTANCE:
graph G, number K
QUESTION:
does G have a clique of size  K?
Subset-Sum
INSTANCE:
numbers a1,...,an,B
QUESTIONS:
is there S {1,...,n} such that
a =B
iS
i
3-COL
INSTANCE:
graph G
QUESTION:
can the vertices of G be assigned
colors red,green,blue so that
no two neighboring vertices have
the same color?
3-SAT  3-COL
R
x
G
x
B
B
G=true
y
x
G
G
G
z
G
R
Planar-3-COL
INSTANCE:
planar graph G
QUESTION:
can the vertices of G be assigned
colors red,green,blue so that
no two neighboring vertices have
the same color?
3-COL  Planar-3-COL
4-COL
INSTANCE:
graph G
QUESTION:
can the vertices of G be assigned
one of 4 colors so that no two
neighboring vertices have the
same color?
3-COL  4-COL
3-COL  4-COL
G

G
planar 4-COL
INSTANCE:
planar graph G
QUESTION:
can the vertices of G be assigned
one of 4 colors so that no two
neighboring vertices have the
same color?
planar 3-COL  planar 4-COL ???
4-COL  3-COL
4-COL  NP
Cook  4-COL  3-SAT
3-SAT 3-COL
Thus:
4-COL  3-COL
2-COL  3-COL
2-COL  3-COL
G

G
3-COL  2-COL ???
2-COL in P