JJ Advanced Complexity Theory April

Lecture

Lecturer Dan Spielman Scribe Ronnie Misra

Recap

Last time we discussed the class of ArthurMerln games Recall that

Denition L AMkn if a PTIME verier A and a polynomial pn st

2

x L a pnprover P st PrA P accepts

3

1

x L pnprovers P PrA P accepts

3

In this lecture we will show some applications of IP and discuss the class PCP

Graph Isomorphism

Although there are no known PTIME algorithms for graph isomorphism we can show that ISO is

probably not NPcomplete

P P

Theorem If ISO is NPcomplete then

3 3

Pro of In the problem set we showed that MA AM proved as NP BP P BP NP P

In fact the AM sp eedup theorem says that

k (n)

AMkn AM

2

AMk AM constants k

MAM AM

We also showed in the problem set that AM BP NP P NPp oly

ISO is NPcomplete

NISO is coNPcomplete

coNP MAM

coNP AM

coNP NPp oly

P P

3 3

Probabilistically checkable pro ofs

We would like to describ e a tough verier that accepts a pro of that can b e sp ot checked

P

Denition A Rn Qn verier is a probabilistic PTIME oracle TM M that

gets Rn random bits

is limited to Qn bits from its oracle P

Denition L PCPRn Qn if a Rn Qn verier V st

�

x accepts x L st PrV

1

�

x L PrV x accepts

2

Here is the probabilistically checkable pro of PCP Note that we dont really think of as

an oracle Instead we are using OTM notation just as a way to show that V gets access to random

bits of

Theorem NEXP PCPPolyn Polyn

Even though pro ofs for languages in NEXP are exp onentially long randomized access allows

such pro ofs to b e veried in PTIME

Theorem NP PCPOlog n O

As a consequence we can use PCP to show that some approximation problems are NPhard

SAT approximation is NPhard

SAT NP PCPOlog n O

a Olog n O verier for SAT

Assume V is nonadaptive or that it ips R Olog n coins then queries the pro of at Q O

places determined only by these random bits If V is adaptive we can construct a nonadaptive

version this version may use exp onentially more queries but this is still a constant

R

On input assume V gets random bits r f g Let q q denote the indices of the

r1 rQ

V reads Q bits of that V queries Thus if

1 2 N q q

r1 r Q

R

construct a small SAT instance on inputs and some For each r f g

r q q

r1 rQ

auxilliary bits Assume has at most S clauses Since Q O S O should b e satised

r r

have values that would cause V to accept when

q q

r1 rQ

Let

r

R

r f01g

is p olynomial in the size of and has inputs as well as all the auxilliary bits

q q

r1 rQ

also has some sp ecial prop erties

R

SAT SAT since V will accept for an y r f g

1

SAT P r V rejects

2

R

r f01g

1

at most of the clauses are satisable from aux bits

r

2

1

aux there is at least one unsatised clause in at least of the clauses

r

2

1

at least a fraction of the clauses of are unsatisable

2S

1

the maximum fraction of satisable clauses of is approximating SAT

2S

1

within a factor of is NPhard where S is some constant dep endant on Q

4S