Ch8 NPC
Millennium Prize Problems
http://en.wikipedia.org/wiki/Millennium_Prize_Problems
8 NP-complete problem
Hard problems: demo
NP-hard (Non-deterministic Polynomial-time hard), in computational complexity theory, is
a class of problems that are, informally, "at least ashard as the hardest problems in NP".
NPC
Informal Definition of NP

Solutions to the problem can be verified easily (in
polynomial time).

The idea of “guessing” the right answer or checking all
possible solutions in parallel to determine which is
correct is called non-determinism.

Any problem with an algorithm that runs on a nondeterministic machine in polynomial time is called a
problem in NP.

Is Towers of Hanoi problem in NP?
2-Coloring
2-coloring (2Col): Given a graph G, can each of its
vertices be labeled with one of 2 different “colors”,
such that no two adjacent vertices have the same label?




Is 2-coloring problem in NP ?
Yes
 Determine the graph is a
Is it in P?
bipartite
graph or not
Yes
1
3-Coloring
3-Coloring
3-coloring (3Col): Given a graph G, can each of its
vertices be labeled with one of 3 different “colors”,
such that no two adjacent vertices have the same label?
3-Coloring



3-Coloring
Is 3-coloring problem in NP ?
Yes
Is it in P?



Hamilton cycle problem
4-coloring

Four color theorem – a planar map can be
colored properly using 4 colors.
The conjecture was first proposed in 1852
 1970s, was proved with help from computer
programs
Is 3-coloring problem in NP ?
Yes
It is in NPC

Hamilton cycle



Hamilton cycle problem:


a simple cycle contains all vertices
Does a given undirected graph have a Hamilton
cycle ?
Is Hamilton cycle problem in NP?

Yes
2
Vertex Cover
Traveling salesman problem

Given a list of cities and the distances between
each pair of cities, what is the shortest possible
route that visits each city exactly once and
returns to the origin city?


Given a graph G, a vertex cover for G is a subset C of
the vertices such that, for every edge (u,v) of G, uC or
vC. The optimization goal is to find as small a vertex
cover for G as possible.
Vertex cover problem: Given a graph G and an
integer k, determine whether there is a vertex cover for
G containing at most k vertices.
B
B
A
D
D
C
Vertex Cover
2SAT

Often expressed as 2CNF (conjunctive normal form - a
conjunction of clauses, where a clause is a disjunction
of literals and there are n different literals)
Ex:
(xy)(yz)(xz)(zy)


Is vertex cover problem in NP ?
Yes
The 2-satisfiability problem is to find a truth
assignment to these variables that makes a
formula of this type true. It’s a decision
problem.
exercise
(x  y) (xz)
kSAT
Maximal number of
literals per clause
P
NP-hard
(xy)(yz)(xz)(zy)
4
3
2
1
3
2SAT is in P
Graph Construction
Theorem: 2SAT is polynomial-time decidable.

Vertex for each variable and a negation of the
variable
Proof: using path searches (DFS/BFS) in
graphs…

Edge (,) iff there exists a clause equivalent to
()
Graph Construction: Example
(xy)(yz)(xz)(zy)
Observation
Claim: If the graph contains a path from  to , it
also contains a path from  to .
More edges?
Note: xy  yx
x
y
x
Proof: If there’s an edge (,), then there’s also an
edge (,).
y
z
z
Corollary
A 2-CNF formula  is unsatisfiable iff there
exists a variable x, such that:
 there is a path from x to x in the graph
AND
 there is a path from x to x in the graph
Graph Construction: Example
(xy)(yz)(xz)(zy)(zx)
x
y
x
y
z
z
4
IF


Only IF
Suppose there are paths x..x and x..x for
some variable x, AND there’s also a satisfying
assignment .
If (x)=T (similarly for (x)=F):
() =?
() is false!


Suppose there are no such paths.
Construct an assignment as follows:
1. pick an
unassigned literal ,
with no path from 
to , and assign it
T
2. assign T to all
reachable vertices
x
yy
xx
x
T
. . .
T

x
F
F
y
z
zz
Only IF (2)
Claim: The algorithm is well defined.
Proof: If there were a path from x to both y
and y, then there would have been a path
from x to y and from y to x which
indicate a path from x to x.
3. assign F to their
negations
4. Repeat until all vertices are
assigned
Corollary
Claim: a 2-CNF formula  is unsatisfiable iff
there exists a variable x, such that:
 there is a path from x to x in the graph
AND
 there is a path from x to x in the graph
3SAT
2SAT is in P
We get the following efficient algorithm for 2SAT:
For each variable x find if there is a path from x to
x and vice-versa.
 Reject if any of these tests succeeded.
 Accept otherwise

 2SATP.

Is E satisfiable?

3SAT is in NP – a decision problem that can be
solved in nondeterministic polynomial time
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