1. P versus NP a Millennium Prize Problem
Chung-Chih Li
School of Information Technology, ISU
March 21, 2018
STV 346
11/23/2018
Math Talk

2. 7 Millennium Problems Clay Mathematics Institute
Yang–Mills and Mass Gap
Riemann Hypothesis
P vs NP Problem
Navier–Stokes Equation
Hodge Conjecture
Poincaré Conjecture (solved, Grigori Perelman/Richard Hamilton)
Birch and Swinnerton-Dyer Conjecture
11/23/2018
Math Talk

3. The hardness of a problem  whether an efficient algorithm exists
Binary Search, O(log n), supper fast
Sequential Search, O(n), fast
Sorting, O(n log n), good
Shortest Path, O(n2), OK
Minimum spanning tree, O(n2), OK
Maximum flow, not so bad
Euler Path, not so bad
Clique, not so good
Hamiltonian path, not good
Vertex cover, feel bad,
SAT, bad
…..
What is going on?
11/23/2018
Math Talk

4. Boolean formula: f(a,b,c)
Satifibility (SAT): is a given Boolean formula satifiable?
Boolean formula: f(a,b,c) a
a = F
a = T b b
b = F
b = T
b = F
b = T c c c c
c = F
c = T
c = F
c = T
c = F
c = T
c = F
c = T F T T F F F F F
 f(a,b,c)
11/23/2018
Math Talk

5. Boolean formula: f(a,b,c)
Backtracking algorithm a
a = F
a = T
Exp b b
b = F
b = T
b = F
b = T c c c c
c = F
c = T
c = F
c = T
c = T
c = F
c = T
c = F F T T F F F F F
 f(a,b,c)
11/23/2018
Math Talk

6. P << Exp
11/23/2018
Math Talk

7. Boolean formula: f(a,b,c)
Nondeterministic algorithm
Boolean formula: f(a,b,c) a
a = F
a = T b b
b = F
b = T
b = F
b = T N P c c c c
c = F
c = T
c = F
c = T
c = F
c = T
c = F
c = T F T T F F F F F
If one of the result is T, then f is satisfiable.
f(a,b,c)
11/23/2018
Math Talk

8. Nondeterministic algorithms are not randomized algorithms.
Nondeterministic algorithms can’t be implemented using parallel computers.
Nondeterministic algorithms can’t be implemented using quantum computers. a
a = F
a = T b b
b = T
b = F
b = F
b = T c c c c
c = F
c = T
c = F
c = T
c = F
c = T
c = T
c = F F T T F F F F F
f(a,b,c)
11/23/2018
Math Talk

9. Is there a 3-clique? NP v1 v2 v5 v4 v3 11/23/2018 Math Talk v1v2v3

10. p Is there a 3-clique? Is there a 2-vertex-cover?
 complement graph 
this can be done in P
11/23/2018
Math Talk

11. Is there A  S, such that  A = 2730 ?
Is there a 2-vertex-cover?
S = {1, 4, 16, 64, 256, 1040, 1089, 1108, 1280, 1285 } v1
Is there A  S, such that  A = ? v5 v2 v4 v3
A = {1, 16, 64, 256, 1108, 1285}
How about  A = ?
11/23/2018
Math Talk

12. Is there a 2-vertex-cover?45 44 43 42 41 40 e1 e2 e3 e4 e5 v1 1
1280 v2
1108 v3
1040 v4
1285 v5
1089
256 64 16 4 2
2730 v1 e1 e2 v5 v2 e5 e3 e4 v4 v3
11/23/2018
Math Talk

13. Is there a 9-vertex-cover?
(x  y  z)  (x   y  z)  ( x   y   z)
Is there a 9-vertex-cover? x x y z z y y x z z z z
11/23/2018
Math Talk

14. Cook Reduction, Polynomial Time Reduction
A  p B
A is reduced to B
B reduces A
SAT  p Vertex-Cover
Clique  p Vertex-Cover
Clique p Vertex-Cover
Vertex-Cover  p Clique
11/23/2018
Math Talk

15. Problem A is called NP-complete if A is NP
Every NP problem can be reduced to A
Steven Cook (1971): SAT is an NP-Complete Problem.
11/23/2018
Math Talk