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42
Parallel repetition: Simplifications and the nosignaling case
 In STOC’07
, 2007
"... Consider a game where a referee chooses (x,y) according to a publicly known distribution PXY, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value a and Bob responds with a value b. Alice and Bob jointly win if a publicly known predicate Q(x,y, a, b) hol ..."
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Cited by 76 (0 self)
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Consider a game where a referee chooses (x,y) according to a publicly known distribution PXY, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value a and Bob responds with a value b. Alice and Bob jointly win if a publicly known predicate Q(x,y, a, b) holds. Let such a game be given and assume that the maximum probability that Alice and Bob can win is v < 1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most ¯v log(s), where s is the maximal number of possible responses from Alice and Bob in the initial game, and ¯v < 1 is a constant depending only on v. In this work, we simplify Raz’s proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication among them. 1
Two Query PCP with SubConstant Error
, 2008
"... We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."
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Cited by 57 (6 self)
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We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with twoquery projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with subconstant error and almostlinear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almostlinear reductions. Previously, the best known N Phardness
A counterexample to strong parallel repetition
 In Proc. 49th FOCS. IEEE
, 2008
"... The parallel repetition theorem states that for any twoprover game, with value 1 − ɛ (for, say, ɛ ≤ 1/2), the value of the game repeated in parallel n times is at most (1 − ɛ c) Ω(n/s) , where s is the answers ’ length (of the original game) and c is a universal constant [R95]. Several researchers ..."
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Cited by 39 (2 self)
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The parallel repetition theorem states that for any twoprover game, with value 1 − ɛ (for, say, ɛ ≤ 1/2), the value of the game repeated in parallel n times is at most (1 − ɛ c) Ω(n/s) , where s is the answers ’ length (of the original game) and c is a universal constant [R95]. Several researchers asked wether this bound could be improved to (1 − ɛ) Ω(n/s) ; this question is usually referred to as the strong parallel repetition problem. We show that the answer for this question is negative. More precisely, we consider the odd cycle game of size m; a twoprover game with value 1 − 1/2m. We show that the value of the odd cycle game repeated in parallel n times is at least 1 − (1/m) · O ( √ n). This implies that for large enough n (say, n ≥ Ω(m 2)), the value of the odd cycle game repeated in parallel n times is at least (1 − 1/4m 2) O(n). Thus: 1. For parallel repetition of general games: the bounds of (1−ɛ c) Ω(n/s) given in [R95, Hol07] are of the right form, up to determining the exact value of the constant c ≥ 2. 2. For parallel repetition of XOR games, unique games and projection games: the bounds of (1 − ɛ 2) Ω(n) given in [FKO07] (for XOR games) and in [Rao07] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game: the bound of 1 − (1/m) · ˜ Ω ( √ n) given in [FKO07] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of MaxCut that are at least 1 − ɛ 2 satisfiable from instances that are at most 1 − (2/π) · ɛ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition. 1
Unique Games on Expanding Constraint Graphs are Easy (Extended ABstract)
 STOC'08
, 2008
"... We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander. We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique ga ..."
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Cited by 39 (11 self)
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We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander. We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.
Information Equals Amortized Communication
, 2010
"... We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, sothat the expected number of bits communicated in the simulationis closeto the amount ofadditionalinformationthatthemessagerevealstothereceiver. Thisisageneralizationandstreng ..."
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Cited by 38 (6 self)
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We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, sothat the expected number of bits communicated in the simulationis closeto the amount ofadditionalinformationthatthemessagerevealstothereceiver. Thisisageneralizationandstrengtheningof the SlepianWolftheorem, which showshow to carryout such a simulation with low amortized communication in the case that M is a deterministic function of X. A caveat is that our simulation is interactive. As a consequence, we obtain new relationships between the randomized amortized communication complexity of a function, and its information complexity. We prove that for any given distribution on inputs, the internal information cost (namely the information revealed to the parties) involved in computing any relation or function using a two party interactive protocol is exactly equal to the amortized communication complexity of computing independent copies of the same relation or function. Here by amortized communication complexity we mean the average per copy communication in the best protocol for computing multiple copies, with a bound on the error in each copy. This significantly simplifies the relationships between the various measures of complexity for average case communication protocols, and proves that if a function’s information cost is smaller than its communication complexity, then multiple copies of the function can be computed more
Approximation Resistant Predicates From Pairwise Independence
, 2008
"... We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwis ..."
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Cited by 32 (5 self)
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We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q] k whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that • For general k ≥ 3 and q ≥ 2, theMAX kCSPq problem is UGhard to approximate within O(kq 2)/q k + ɛ. • For the special case of q =2, i.e., boolean variables, we can sharpen this bound to (k + O(k 0.525))/2 k + ɛ, improving upon the best previous bound of 2k/2 k +ɛ (Samorodnitsky and Trevisan, STOC’06) by essentially a factor 2. • Finally, again for q =2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the O(k 0.525) term can be replaced by the constant 4. 1
Unbounded violations of bipartite Bell inequalities via Operator Space theory
 Communications in Mathematical Physics, 300(3):715–739, 2010. arXiv:0910.4228. Shorter version appeared in PRL 104:170405, arXiv:0912.1941
"... Abstract: In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order Ω n log2 n when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator sp ..."
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Cited by 17 (2 self)
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Abstract: In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order Ω n log2 n when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative Lp embedding theory. As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise. 1.
Parallel Repetition of Entangled Games ∗
"... We consider oneround games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of provers? Classically, th ..."
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Cited by 16 (3 self)
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We consider oneround games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of provers? Classically, this question, open for many years, has culminated in Raz’s celebrated parallel repetition theorem on one hand, and in efficient product testers for PCPs on the other. In the case where provers share entanglement, the only previously known results are for special cases of games, and are based on techniques that seem inherently limited. Here we show for the first time that the maximum success probability of entangled games can be reduced through parallel repetition, provided it was not initially 1. Our proof is inspired by a seminal result of Feige and Kilian in the context of classical twoprover oneround interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest. Twoprover games play a major role both in theoretical computer science, where they led to many breakthroughs such as the discovery of tight inapproximability results, and in quantum physics, where they first arose in the context of Bell inequalities. In such games, a referee chooses a pair of questions
Rounding Parallel Repetitions of Unique Games
"... We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically, denoting by val(G) the value of a twoprover unique game G, and by sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every ..."
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Cited by 15 (5 self)
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We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically, denoting by val(G) the value of a twoprover unique game G, and by sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every ℓ ∈ N, if sdpval(G) � 1 − δ, then val(G ℓ) � 1 − √ sℓδ. Here, G ℓ denotes the ℓfold parallel repetition of G, and s = O(log ( k/δ)), where k denotes the alphabet size of the game. For the special case where G is an XOR game (i.e., k = 2), we obtain the same bound but with s as an absolute constant. Our bounds on s are optimal up to a factor of O(log ( 1/δ)). For games with a significant gap between the quantities val(G) and sdpval(G), our result implies that val(G ℓ) may be much larger than val(G) ℓ, giving a counterexample to the strong parallel repetition conjecture. In a recent breakthrough, Raz (FOCS ’08) has shown such an example using the maxcut game on odd cycles. Our results are based on a generalization of his techniques.
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1