AM With Multiple Merlins Scott Aaronson MIT Scott Aaronson (MIT) Dana Moshkovitz (MIT) Russell Impagliazzo (UCSD)

AM With Multiple Merlins

Scott AaronsonMIT

Scott Aaronson (MIT)

Dana Moshkovitz (MIT)

Russell Impagliazzo (UCSD)

Two-Prover Games(the first slide of, like, half of all complexity talks)

ArthurMerlin1Merlin2

ybxayxVEGDyxBYbAXa

,,,max:~,:,:

xX yY

a(x)A b(y)B

Dyx ~,

The PCP Theorem: Given G=(X,Y,A,B,D,V), it’s NP-hard even just to decide whether (G)=1 or (G)<0.01

1,0,,, bayxV

The “Scaled-Up” Version [BFL’91]: MIP = NEXP

“VALUE” OF THE GAME (WHAT THE MERLINS ARE TRYING TO MAXIMIZE):

This work: What if the challenges to the Merlins have to be independent?

“Free Games”: G’s for which D is a product distributionOr for simplicity, let’s say, the uniform distribution

Obvious Objection: The whole power of MIP comes from Arthur’s ability to correlate questions—take that away, and two-prover games should become trivial!

As we’ll see, that’s not entirely true…

AM(2): Complexity class based on free games. Two-prover, one-round MIP, but where Arthur’s challenges to the two non-communicating Merlins have to be independent, uniform random strings

A known concept in PCP. Yet we seem to be the first to explicitly study the complexity of free games

Summary of ResultsResult #1: There’s an AM(2) protocol by which Arthur can become convinced that a 3SAT instance of size n is satisfiable, by sending just Õ(n) random bits to the Merlins, and getting back Õ(n)-bit answers

Result #2: Given a free game G of size n, there’s analgorithm to approximate (G) within nOn log2

Assuming the ETH, both of these results imply the other’s near-

optimality!

3SAT

instance Free game G of size

nO2

Can approximate (G) (and thereby decide

) in time

nOnO

nO

222log

Which means that, assuming 3SAT requires 2Ω(n) time:

• AM(2) protocols for 3SAT need communication

• Approximating free games requires time

• Approximating dense CSPs with polynomial-size alphabets also requires time[Barak et al. 2011] gave an nO(log n)-time algorithm for such CSPs, but its running time was never previously explained

n~

nn log

~

nn log~

Going FurtherOur algorithm for free games implies AM(2) EXP—improving on the trivial bound AM(2) MIP = NEXP

But AM AM(2) EXP is still quite a gap!

Result #3: AM(2) = AM(with an inherent quadratic blowup in communication)

And more generally, AM(k) = AM for all k=poly(n)

Proof relies heavily on previous work on dense CSPs: [Alon et al. 2002], [Barak et al. 2011]

Result #1: 3SAT ProtocolLet be a 3SAT instance of size n. Can assume w.l.o.g. that is a balanced PCP, with only polylog blowup [Dinur 2006]

Standard “Clause/Variable Game”:

Random clause C

CHECKS SATISFACTION & CONSISTENCY

Random variable xC

Assignment to C Assignment to x

“Birthday Game”:

Clauses C1,…,CK Variables x1,…,xL

Assignments to C1,…,CK Assignments to x1,…,xL

CHECKS SATISFACTION & CONSISTENCY ON BIRTHDAY COLLISIONS

nLK ,

Proving The 3SAT Protocol SoundSuppose the Merlins can cheat in the “birthday game.” We show how they can also cheat in the original clause/variable game, thereby giving a contradiction

Clause C Variable xC

“Smuggles” C among random clauses C1,…,CK that he picks

himself

“Smuggles” x among random variables x1,

…,xL that he picks himself

Then the Merlins run their birthday strategy on C1,…,CK and x1,…,xL, and return the results restricted to C and x

Key Technical Claim (proved with second-moment method): The induced distribution over C1,…,CK and x1,…,xL is -close in variation distance to the uniform distribution And then we’re done!

High-Error Case: If we only want a 1 vs. 1- soundness gap, a different argument gives an AM(2) protocol for 3SAT with communication.Hence, assuming ETH, deciding whether a free game G satisfies (G)=1 or (G)<1- requires time

KL

nO

nnO polylog

nn log1~

Low-Error Case: If we want a 1 vs. gap, switching from [Dinur 2006] to [Moshkovitz-Raz 2008] gives an AM(2) protocol for 3SAT with communication.Hence, assuming ETH, deciding whether a free game G satisfies (G)=1 or (G)< requires time

/1poly12/1 on

11logpoly onn

Result #2: Approximation Algorithm for Free Games

Algorithm’s Running Time:

yY

xX

Let v be the value of the best pair of strategies that this algorithm finds

Clearly v(G)

Furthermore, v(G)- w.h.p. over S, by union and Chernoff bounds

S

2

log

BY

OS

nOSnYAXAO log2

Can derandomize by looping over all possible S

Best responses

Best

resp

onse

s

Loop over all possible strategies on S

Followup Work [Brandão-Harrow]: A different algorithm for

approximating free games, with exactly the same running time as ours, but based on LP relaxation

Result #3: AM(2) = AMSubsampling Theorem: Let G be any free game, and let GS,T be the subgame induced by restricting Merlin1’s challenges to SX and Merlin2’s to TY, where |S|=|T|=log(|A||B|)/O(1). Then

yY

xX

GGEG TSTS

,,

S

T

Trivial Not Trivial(but [Alon et al. 2002],

[Barak et al. 2011] already did most of the work)

The AM simulation of an AM(2) protocol is then simply: Arthur chooses S,T, then Merlin replies with a:SA, b:TB, then Arthur verifies that (GS,T) is large

Generalizing to k Merlins

Can’t we do better, by encoding free games as dense CSPs? Alas, straightforward encoding fails when k=(log n)

We find a better encoding, which yields: (1) AM(k) = AM for all k=poly(n), and (2) any AM(k) protocol for 3SAT needs total communication (assuming ETH)

Let G be a k-player free game (k3). By applying our two-player algorithm recursively, to “peel off Merlins one at a time,” we can approximate (G) to within in time

nkOn log22

This implies (1) AM(k) EXP, and (2) any AM(k) protocol for 3SAT needs communication assuming the ETH 4/1~

n

n~

Quantum MotivationQMA(2): Arthur receives two unentangled quantum proofs, |1 from Merlin1 and|2 from Merlin2

Best current knowledge: QMA QMA(2) NEXP. Pathetic!

[ABDFS, CCC’2008]: There’s a QMA(2) protocol to prove that a 3SAT instance of size n is satisfiable, using quantum messages with Õ(n) qubits onlyProtocol uses PCP Theorem and Birthday Paradox in almost exactly the same way as our AM(2) protocol!

Conjectures: QMA(2) EXP. The square-root savings of [ABDFS’2008] is optimal, assuming the ETH.

Upshot of This Work: Everything we’d like to

prove about QMA(2), we can prove about AM(2)!

Slide Where I Try to Provoke You

Think about it: we gave an Õ(n)-communication AM(2) protocol for 3SAT, and an nO(log n) approximation algorithm for free games. Neither result “knew about the other.” Yet, if either had been slightly better, their combination would’ve falsified ETH. So if ETH is false, how did the two results “coordinate”?

Should one call results like ours “evidence” for the ETH?

Open ProblemsÕ O? 1/2 1/?

Birthday Repetition Theorem?

Is our 3SAT protocol non-algebrizing?It’s definitely non-relativizing

Better approximation algorithms for free projection games and free unique games?

Conjecture: Exists a PTAS but not an FPTAS

AM(2) with entangled provers?

Use our techniques to show nΩ(log n) hardness for approximate Nash equilibrium, assuming ETH?

[Hazan-Krauthgamer 2009]: assuming planted clique is hard