CSE 565 , lecture 17 , 10/25/2021
tact : P C- Np .
Question Pt NP I open , consensus opinion : PHP)
.
Fact : Let ✗ c- NPC . It ✗ EP , then p=Np .
Proof : Let X
‘
be an arbitrary problem in NP
–
✗ c- Npc ⇒ ✗
‘
Ep X . ⇒ YEP 1 because ✗ c- P)
⇒ NPEP ⇒ p=NP ☒ .
Hi⇒ : p ⇒ NP
⑤ ↳NpnNpc=0_
p=NP=NPc NP
Def_lw_¥ problem) : Let ✗ be a decision problem .
It’s complementary problem , denoted as I , is defined as
the same set instances with X but with opposite ground truth .
I 😐
¥÷¥:÷
a
– a
–
r a
✗ ☒
Exampte:
3.SAT ( CNF) 3SAT
true : ← 7- assignment sit . CNF is true . → false
false :-c V- assignment sit CNF is false . → true
=a¥ I =X
Fonti XEP ⇐ I c- P
Question it ✗ c- NP ⇒ I c- NP
Det ko-µ :p) : Let ✗ be a decision problem . We say XC-co-NP.it
there exists a polynomial – time procedure A aerified , sit .
i. input of A is an instance ✗ of ✗ plus %%ÉÑiticate c.^
Isolation
2. For any instance × of ✗
if ✗ is a
”
false
”
– instance
,
then 7- c. sit . Alx . =
”
no
”
if ✗ is a “true
”
– instance
,
then V-c.s.t-AH.cl =
”
yes
”
.
Faet ✗ c- NP it and only if I c- co-NP
.
proofs by the defs of co-NP , NP , and complementary problem .
Fanti PE co- NP
Inti P E NP A co
-Np
–
Question : NP ?= co – Np . Copen . consensus : Np + co -Np)
=ae# If p=NP , then NP = co -Np .
proof ; let ✗ c- Np .⇒ ✗ c-P ⇒ Iep ⇒ IENP ⇒ ☒ c- co
-Np⇒ ✗ c- warp.
let ✗ c- co – Np ⇒ IENP ⇒ IEP ⇒ XEP ⇒ ✗ c-NP ☒ .
Fa It NP # co – NP . then P # Np .
Hierarchy : p # NP
← ↳
NP ?= co -NP
0
P=Np=w-µp * ↳
P=? NPN co- Np
.
yes/ ↳⑤
④ ④NP=co-NP
Np co -NP NP a-NP
Question : P ?= NPA co -NP Copen : mixed)
problem X: Does a G-ME ) contains a flow with 1ft > N ?
✗ c- NP .arhfiaatef-XC-co-NP.mn/-f1owf*–.s-twtlS.T) as certificate .
false instance ⇒ 7-1 f- sit . Ifl > N
⇐ V-fsit.lt/sN-1 .
⇐ * ISN -1
⇐ 454T¥ a- N – I
function verifier 16–14 El . N, )
verify Is , -11 is an s -t cut .
I verity 451) IN -1
end .
Probtem ( integer factorization) .
Input an integer X, an integer n .
output’s it there exists a factor of ✗ ( ✗ mod ✗ ⇒ sit . ✗en .
Algo – for
– IF IX. n)
for K= 2 to n
1 it ✗
mod K=o return true1 end
end .
return false
Running time
i Qln)_= -012109¥ exponential- time
siu-tip-uillog.lt/ogzn)
Question : IF & P I open) .
FAI IF c- NP
proot : certificate ✗ lfaetor)
verifier IX. n , x)
verity ✗ enI verity ✗ mod ✗ =0
end ☒ .
Claim : IF c- co – NP
‘
Fae false instance ( Xin) of IF
⇐ V- factor ✗ of ✗ . sit . × > n .
⇐ it prime factor X of X , sit . X > n .
Exampte : 11=7×11×133 W= 6 , only need to check {7.11.13}
.
✗ =p ,M . pztkpgn} . . . prank ( facto’t:-. ration)
K
toga = nilogp , 1- nztogzpz
+ – – – nk.bg?k–E,ni.logzP;
2
Pi > 2 , ni> 1 .
109N = ,É
,
ni.log.Pi-3-i.kz
,
ni
. ⇒ K= 0110g X) .
¥iat : ( Pnh ,) , 1Pa, nd . ‘
‘ ‘
, CPK, NK) .
function verifier IX. n . { cpi.mil/1–i–k}) .
verity ✗ = II
,
Pini .
| verity Pi > n . KKK
end
veritypiis-apr.me. kick
T-ati-PRIMES-ltodediide.it a number is a prime. c- P –
l AKS
, 20021 .
Fonti Above verifier runs in poly -time .
Eatin : ✗ d- IPP ?