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

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Two-Prover Games (the first slide of, like, half of all complexity talks) Arthur Merlin 1 Merlin 2 xXxX yYyY a(x) A b(y) B 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 “VALUE” OF THE GAME (WHAT THE MERLINS ARE TRYING TO MAXIMIZE):

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This work: What if the challenges to the Merlins have to be independent? “Free Games”: G’s for which D is a product distribution Or 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

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Summary of Results Result #1: There’s an AM(2) protocol by which Arthur can become convinced that a 3S AT 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 an algorithm to approximate (G) within Assuming the ETH, both of these results imply the other’s near- optimality!

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3S AT instance Free game G of size Can approximate (G) (and thereby decide ) in time Which means that, assuming 3S AT requires 2 Ω(n) time: AM(2) protocols for 3S AT need communication Approximating free games requires time Approximating dense CSPs with polynomial-size alphabets also requires time [Barak et al. 2011] gave an n O(log n) -time algorithm for such CSPs, but its running time was never previously explained

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Going Further Our 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]

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Result #1: 3S AT Protocol Let 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 x C Assignment to C Assignment to x “Birthday Game”: Clauses C 1,…,C K Variables x 1,…,x L Assignments to C 1,…,C K Assignments to x 1,…,x L CHECKS SATISFACTION & CONSISTENCY ON BIRTHDAY COLLISIONS

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Proving The 3S AT 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 Clause C Variable x C “Smuggles” C among random clauses C 1,…,C K that he picks himself “Smuggles” x among random variables x 1,…,x L that he picks himself Then the Merlins run their birthday strategy on C 1,…,C K and x 1,…,x L, and return the results restricted to C and x

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Key Technical Claim (proved with second-moment method) : The induced distribution over C 1,…,C K and x 1,…,x L 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 3S AT 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 3S AT with communication. Hence, assuming ETH, deciding whether a free game G satisfies (G)=1 or (G)< requires time

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Result #2: Approximation Algorithm for Free Games Algorithm’s Running Time: yYyY xXxX 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 Can derandomize by looping over all possible S Best responses 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

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Result #3: AM(2) = AM Subsampling Theorem: Let G be any free game, and let G S,T be the subgame induced by restricting Merlin 1 ’s challenges to S X and Merlin 2 ’s to T Y, where |S|=|T|=log(|A||B|)/ O(1). Then yYyY xXxX 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:S A, b:T B, then Arthur verifies that (G S,T ) is large

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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 3S AT needs total communication (assuming ETH) 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 3S AT needs communication assuming the ETH

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Quantum Motivation QMA(2): Arthur receives two unentangled quantum proofs, | 1 from Merlin 1 and| 2 from Merlin 2 Best current knowledge: QMA QMA(2) NEXP. Pathetic! [ABDFS, CCC’2008]: There’s a QMA(2) protocol to prove that a 3S AT instance of size n is satisfiable, using quantum messages with Õ( n) qubits only Protocol 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)!

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Slide Where I Try to Provoke You Think about it: we gave an Õ( n)-communication AM(2) protocol for 3S AT, and an n O(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?

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Open Problems Õ O? 1/ 2 1/ ? Birthday Repetition Theorem? Is our 3S AT 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 n Ω(log n) hardness for approximate Nash equilibrium, assuming ETH? [Hazan-Krauthgamer 2009]: assuming planted clique is hard

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