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AM With Multiple Merlins

AM With Multiple Merlins. Scott Aaronson MIT. Scott Aaronson (MIT) Dana Moshkovitz (MIT) Russell Impagliazzo (UCSD). Two- Prover Games (the first slide of, like, half of all complexity talks). Arthur. Merlin 2. Merlin 1. y  Y. x  X. b (y)B. a (x)  A.

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AM With Multiple Merlins

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

  2. Two-Prover Games(the first slide of, like, half of all complexity talks) Arthur Merlin2 Merlin1 yY xX b(y)B a(x)A “VALUE” OF THE GAME (WHAT THE MERLINS ARE TRYING TO MAXIMIZE): 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 The “Scaled-Up” Version [BFL’91]:MIP = NEXP

  3. This work: What if the challenges to the Merlinshave to be independent? “Free Games”: G’s for which D is a product distributionOr for simplicity, let’s say, the uniform distribution A known concept in PCP. Yet we seem to be the first to explicitly study the complexity of free games 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 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…

  4. Summary of Results Result #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 Assuming the ETH, both of these results imply the other’s near-optimality! Result #2: Given a free game G of size n, there’s an algorithm to approximate (G) within 

  5. Can approximate (G) (and thereby decide ) in time Free game G of size 3Sat instance  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 alsorequires time[Barak et al. 2011] gave an nO(log n)-time algorithm for such CSPs, but its running time was never previously explained

  6. Going Further Our algorithm for free games implies AM(2)EXP—improving on the trivial bound AM(2) MIP =NEXP But AMAM(2)EXPis 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]

  7. Result #1: 3Sat Protocol Let  be a 3SAT instance of size n. Can assume w.l.o.g. that is a balanced PCP, with only polylogblowup[Dinur 2006] Standard “Clause/Variable Game”: Random variable xC Random clause C CHECKS SATISFACTION & CONSISTENCY Assignment to x Assignment to C “Birthday Game”: Variables x1,…,xL Clauses C1,…,CK CHECKS SATISFACTION & CONSISTENCY ON BIRTHDAY COLLISIONS Assignments to x1,…,xL Assignments to C1,…,CK

  8. Proving The 3Sat Protocol Sound Suppose 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 Variable xC Clause C “Smuggles” x among random variables x1,…,xL that he picks himself “Smuggles” C among random clauses C1,…,CK 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

  9. 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 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

  10. Result #2: Approximation Algorithm for Free Games yY 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 Chernoffbounds Best responses Followup Work [Brandão-Harrow]: A different algorithm for approximating free games, with exactly the same running time as ours, but based on LP relaxation xX Best responses S Loop over all possible strategies on S Algorithm’s Running Time: Can derandomize by looping over all possible S

  11. Result #3: AM(2) = AM yY Subsampling Theorem: Let G be any free game, and let GS,T be the subgame induced by restricting Merlin1’s challenges to SX and Merlin2’s to TY, where |S|=|T|=log(|A||B|)/O(1). Then T xX The AM simulation of an AM(2) protocol is then simply: Arthur chooses S,T, then Merlin replies with a:SA, b:TB, then Arthur verifies that (GS,T) is large S Not Trivial(but [Alon et al. 2002], [Barak et al. 2011] already did most of the work) Trivial

  12. Generalizing to k Merlins Let G be a k-player free game (k3). By applying our two-player algorithm recursively, to “peel off Merlins one at a time,” we can approximate (G) to within  in time This implies (1) AM(k)EXP, and (2) any AM(k) protocol for 3Sat needs communication assuming the ETH 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)

  13. Quantum Motivation QMA(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)!

  14. Slide Where I Try to Provoke You Should one call results like ours “evidence” for the ETH? 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”?

  15. Open Problems Õ  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

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