15-252 LECTURE #9, 3/19/2020

pcpandHARDNESSOFAPPROXIMATIONNP-i.IEN P F def. polythene verifier V

s 't see L ⇒ I y.ly/EpolyllxDstVCx,y) acceptKIEL ⇒ Hy , Vln,y) reject

.

€÷? lthynoartosicdel2=3COLOR .

OEG y =3- coloring ofG

Talibfor invalid claims, @¢y, every proof

will

be rejected .

-

Requires reading pf. in entirety .

hazggoaderltt-dcams.com you getbywith good confidence by jcistispotoheckiyu

y ?

God: Develop robust pooofsystcm / proofwriting former set for false claims,there are bugs in the proof everywhere

.

pcptheorem F finite q Gntger)orfollowing holds ,theNP , F a polythene

randomized

verifier V ..

-Tf KEL, Fy , ly Ispoly HH) St

Vln,yD acceptwith prob. I Gertainhf

- IfkEf L off ,I ykpoly 11×11 ,

V (x , y) rejectswith prod z Yz

furthermore,V only probestreads

q locationsof the proof f .

↳ Crandonly chosen)

xefnf.IT#K.aPAprobabilistically checkable proofGo from readingfull proof to a thy tiny

sample of the Pf

q is independentoh Hel, ly tIn fact , can takeq=⑦

Connection to Approximation-

3.SAT is NP-completeso given a

satisfiable 3SAT formula,there is likely no polythene also

to find a

satisfying assignment.

How about findingan approximately good

satisfying assignment ? Saysatisfies 99%

of the clauses.

Also seemshard

1¥prove suchahardness ?

A"gappoodieag

" reductions

CIRCUITSAT Ep APPROX-35AT

Circuit C 1-formula 0 =GapRedCC)

C satisfiable → of saltstrash

C not satisfiable as of is not 99%}Satisfiable

Cie no assignment gets even 994 ofthe clauses satisfied in f)

Exercise : Such a reduction impliesfrrdey a 991. satisfying

assignment to a

fully satisfiable formula is also hard .

183 83=8 , Agz

BE cgivgivgs) n ←ge Ga

① O Cgivgzvgz) n

Cg , RE VE) 1

( g ,Vgz VJs )

In the usual reduction, you can justviolate one gate of circuit I make

it

satisfiable

PCP Theorem⇐ Existence of sucha gap producingreduction

⇐ approximatelyf sakshi 3ffId)PATH# Hardness of- approximation⇒

GarantSAT c * f-GapRedC E L b 3SATfor

mta

Aarantpcp proof : A purported satisfy're

Finalist of GapRed (C )

Verifier .

Tick a random clause of f- check that it is

true under 5

• Reads 3 bits it

. If CEL,write proper o

&

verifier will surelyaccept ✓

• Tf CEIL , verifierrepeats with

> 1% probability .To reject with prob 343,

Asst repeat 04) times-.

@ .99J I 43Bt queries

⑧na you have hardness ofapproxresult for 3.SAT, can provefurther aim approxinability

"

results is viareductionsEf : 3.SAT Ep CLIQUE

4 ↳ Look>Inspection of the reduction

shows :

>Maximum clique size of G

= Max. # clauses of g tha( can be satisfied←

⇒ Finding a 99% approx cliqueis NP - hard

¥8 Finding a 1% approxclique

is NP-hard